Simplex and Polygon Equations
It is shown that higher Bruhat orders admit a decomposition into a higher Tamari order, the corresponding dual Tamari order, and a ''mixed order''. We describe simplex equations (including the Yang-Baxter equation) as realizations of higher Bruhat orders. Correspondingly, a famil...
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irk-123456789-1471052019-02-14T01:27:00Z Simplex and Polygon Equations Dimakis, A. Müller-Hoissen, F. It is shown that higher Bruhat orders admit a decomposition into a higher Tamari order, the corresponding dual Tamari order, and a ''mixed order''. We describe simplex equations (including the Yang-Baxter equation) as realizations of higher Bruhat orders. Correspondingly, a family of ''polygon equations'' realizes higher Tamari orders. They generalize the well-known pentagon equation. The structure of simplex and polygon equations is visualized in terms of deformations of maximal chains in posets forming 1-skeletons of polyhedra. The decomposition of higher Bruhat orders induces a reduction of the N-simplex equation to the (N+1)-gon equation, its dual, and a compatibility equation. 2015 Article Simplex and Polygon Equations / A. Dimakis, F. Müller-Hoissen // Symmetry, Integrability and Geometry: Methods and Applications. — 2015. — Т. 11. — Бібліогр.: 107 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 06A06; 06A07; 52Bxx; 82B23 DOI:10.3842/SIGMA.2015.042 http://dspace.nbuv.gov.ua/handle/123456789/147105 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України |
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It is shown that higher Bruhat orders admit a decomposition into a higher Tamari order, the corresponding dual Tamari order, and a ''mixed order''. We describe simplex equations (including the Yang-Baxter equation) as realizations of higher Bruhat orders. Correspondingly, a family of ''polygon equations'' realizes higher Tamari orders. They generalize the well-known pentagon equation. The structure of simplex and polygon equations is visualized in terms of deformations of maximal chains in posets forming 1-skeletons of polyhedra. The decomposition of higher Bruhat orders induces a reduction of the N-simplex equation to the (N+1)-gon equation, its dual, and a compatibility equation. |
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Dimakis, A. Müller-Hoissen, F. |
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Dimakis, A. Müller-Hoissen, F. Simplex and Polygon Equations Symmetry, Integrability and Geometry: Methods and Applications |
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Dimakis, A. Müller-Hoissen, F. |
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Dimakis, A. |
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Simplex and Polygon Equations |
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Simplex and Polygon Equations |
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Simplex and Polygon Equations |
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Simplex and Polygon Equations |
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Simplex and Polygon Equations |
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simplex and polygon equations |
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Інститут математики НАН України |
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2015 |
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Simplex and Polygon Equations / A. Dimakis, F. Müller-Hoissen // Symmetry, Integrability and Geometry: Methods and Applications. — 2015. — Т. 11. — Бібліогр.: 107 назв. — англ. |
| series |
Symmetry, Integrability and Geometry: Methods and Applications |
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AT dimakisa simplexandpolygonequations AT mullerhoissenf simplexandpolygonequations |
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2025-07-11T01:21:50Z |
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 11 (2015), 042, 49 pages
Simplex and Polygon Equations
Aristophanes DIMAKIS † and Folkert MÜLLER-HOISSEN ‡
† Department of Financial and Management Engineering, University of the Aegean,
82100 Chios, Greece
E-mail: dimakis@aegean.gr
‡ Max Planck Institute for Dynamics and Self-Organization, 37077 Göttingen, Germany
E-mail: folkert.mueller-hoissen@ds.mpg.de
Received October 23, 2014, in final form May 26, 2015; Published online June 05, 2015
http://dx.doi.org/10.3842/SIGMA.2015.042
Abstract. It is shown that higher Bruhat orders admit a decomposition into a higher
Tamari order, the corresponding dual Tamari order, and a “mixed order”. We describe
simplex equations (including the Yang–Baxter equation) as realizations of higher Bruhat
orders. Correspondingly, a family of “polygon equations” realizes higher Tamari orders.
They generalize the well-known pentagon equation. The structure of simplex and polygon
equations is visualized in terms of deformations of maximal chains in posets forming 1-ske-
letons of polyhedra. The decomposition of higher Bruhat orders induces a reduction of the
N -simplex equation to the (N + 1)-gon equation, its dual, and a compatibility equation.
Key words: higher Bruhat order; higher Tamari order; pentagon equation; simplex equation
2010 Mathematics Subject Classification: 06A06; 06A07; 52Bxx; 82B23
1 Introduction
The famous (quantum) Yang–Baxter equation is
R̂12R̂13R̂23 = R̂23R̂13R̂12,
where R̂ ∈ End(V ⊗ V ), for a vector space V , and boldface indices specify the two factors of
a threefold tensor product on which R̂ acts. This equation plays an important role in exactly
solvable two-dimensional models of statistical mechanics, in the theory of integrable systems,
quantum groups, invariants of knots and three-dimensional manifolds, and conformal field theory
(see, e.g., [15, 22, 34, 47, 48]).
A set-theoretical version of the Yang–Baxter equation considers R̂ as a map R̂ : U×U → U×U ,
where U is a set (not necessarily supplied with further structure). Nontrivial examples of “set-
theoretical solutions” of the Yang–Baxter equation, for which Veselov later introduced the name
Yang–Baxter maps [98, 99], apparently first appeared in [91] (cf. [99]), and Drinfeld’s work [29]
stimulated much interest in this subject. Meanwhile quite a number of examples and studies of
such maps have appeared (see, e.g., [1, 32, 35, 42, 96]).
The Yang–Baxter equation is a member of a family, called simplex equations [18] (also see,
e.g., [39, 68, 74, 75, 77]). The N -simplex equation is an equation imposed on a map R̂ : V ⊗N →
V ⊗N , respectively R̂ : UN → UN for the set-theoretical version. The next-to-Yang–Baxter
equation, the 3-simplex equation,
R̂123R̂145R̂246R̂356 = R̂356R̂246R̂145R̂123,
is also called tetrahedron equation or Zamolodchikov equation. This equation acts on V ⊗6.
A set of tetrahedron equations first appeared as factorization conditions for the S-matrix in
mailto:dimakis@aegean.gr
mailto:folkert.mueller-hoissen@ds.mpg.de
http://dx.doi.org/10.3842/SIGMA.2015.042
2 A. Dimakis and F. Müller-Hoissen
Zamolodchikov’s (2 + 1)-dimensional scattering theory of straight lines (“straight strings”), and
in a related three-dimensional exactly solvable lattice model [105, 106]. This has been inspired
by Baxter’s eight-vertex lattice model [11, 12] and stimulated further important work [13, 14],
also see the survey [95]. Meanwhile the tetrahedron equation has been the subject of many
publications (see, in particular, [20, 21, 43, 59, 68, 70, 76, 89]). An equation of similar structure
as the above 3-simplex equation, but acting on V ⊗4, has been proposed in [39].
In a similar way as the 2-simplex (Yang–Baxter) equation describes a factorization condition
for the scattering matrix of particles in two space-time dimensions [23, 103], as just mentioned,
the 3-simplex equation describes a corresponding condition for straight lines on a plane [105, 106].
Manin and Schechtman [78, 79] looked for what could play the role of the permutation group,
which acts on the particles in the Yang–Baxter case, for the higher simplex equations. They were
led in this way to introduce the higher Bruhat order B(N,n), with positive integers n < N . This
is a partial order on the set of certain equivalence classes of “admissible” permutations of
(
[N ]
n
)
,
which is the set of n-element subsets of [N ] := {1, 2, . . . , N} (see Section 2.1). The N -simplex
equation is directly related to the higher Bruhat order B(N + 1, N − 1).
Let us consider the local Yang–Baxter equation1
L̂12(x)L̂13(y)L̂23(z) = L̂23(z′)L̂13(y′)L̂12(x′),
where the L̂ij depend on variables in such a way that this equation uniquely determines a map
(x, y, z) 7→ (x′, y′, z′). Then this map turns out to be a set-theoretical solution of the tetrahedron
equation. Here we wrote L̂ij instead of R̂ij in order to emphasize that such a “localized”
equation may be regarded as a “Lax system” for the tetrahedron equation, i.e., the latter arises
as a consistency condition of the system. This is a familiar concept in integrable systems theory.
If the variables x, y, z are elements of a (finite-dimensional, real or complex) vector space, and
if the maps L̂ij depend linearly on them, then L̂ij(x) = xaL̂aij , using the summation convention
and expressing x = xaE
a in a basis Ea, a = 1, . . . ,m. In this case the above equation takes the
form
L̂a12L̂b13L̂c23 = R̂abcdef L̂d23L̂e13L̂
f
12,
where the coefficients R̂abcdef are defined by z′dy
′
ex
′
f = xaybzcR̂abcdef . The last system is also known as
the tetrahedral Zamolodchikov algebra (also see [16, 59]). Analogously, there is a Lax system for
the Yang–Baxter equation [96], consisting of 1-simplex equations, which is the Zamolodchikov–
Faddeev algebra [69], and this structure extends to all simplex equations [73, 74, 75, 77]. The
underlying idea of relaxing a system of N -simplex equations in the above way, by introducing an
object R̂, such that consistency imposes the (N + 1)-simplex equation on it, is the “obstruction
method” in [25, 44, 73, 74, 75, 77, 82]. Also see [9, 50, 51, 58, 93] for a formulation in the setting
of 2-categories. Indeed, the obstruction method corresponds to the introduction of laxness
(“laxification” [93]).
An equation of a similar nature as the Yang–Baxter equation is the pentagon equation
T̂12T̂13T̂23 = T̂23T̂12, (1.1)
which appears as the Biedenharn–Elliott identity for Wigner 6j-symbols and Racah coefficients
in the representation theory of the rotation group [19], as an identity for fusion matrices in
conformal field theory [84], as a consistency condition for the associator in quasi-Hopf alge-
bras [27, 28] (also see [3, 4, 10, 33, 36, 40, 41]), as an identity for the Rogers dilogarithm func-
tion [87] and matrix generalizations [53], for the quantum dilogarithm [5, 17, 20, 37, 55, 100],
1In very much the same form, the local Yang–Baxter equation appeared in [88], for example. A natural
generalization is obtained by replacing the three appearances of L̂ by three different maps.
Simplex and Polygon Equations 3
and in various other contexts (see, e.g., [26, 52, 56, 57, 60, 67, 72]). In particular, it is satisfied
by the Kac–Takesaki operator (T̂ f)(g, g′) = f(gg′, g′), g, g′ ∈ G, G a group, where it expresses
the associativity of the group operation (see, e.g., [101]). A unitary operator acting on H⊗H,
where H is a Hilbert space, and satisfying the pentagon equation, has been termed a multi-
plicative unitary [6, 7, 8, 46, 81, 90, 101, 104]. It plays an essential role in the development of
harmonic analysis on quantum groups. Under certain additional conditions, such an operator
can be used to construct a quantum group on the C∗ algebra level [6, 45, 90, 101].2 For any
locally compact quantum group, a multiplicative unitary can be constructed in terms of the
coproduct. Any finite-dimensional Hopf algebra is characterized by an invertible solution of
the pentagon equation [83]. The pentagon equation arises as a 3-cocycle condition in Lie group
cohomology and also in a category-theoretical framework (see, e.g., [94]).
A pentagon relation arises from “laxing” the associativity law [71, 92, 97]. In its most basic
form, it describes a partial order (on a set of five elements), which is the simplest Tamari lattice
(also see [85]). This is T (5, 3) in the notation of this work (also see [25]).
We will show that the pentagon equation belongs to an infinite family of equations, which
we call polygon equations. They are associated with higher Tamari orders, as defined3 in [25],
in very much the same way as the simplex equations are associated with higher Bruhat orders
(also see [25]). We believe that these higher Tamari orders coincide with higher Stasheff–Tamari
orders, defined in terms of triangulations of cyclic polytopes [30, 49, 86].4
In Section 2.2 we show that any higher Bruhat order can be decomposed into a correspond-
ing higher Tamari order, its dual (which is the reversed Tamari order), and a “mixed order”.
A certain projection of higher Bruhat to higher (Stasheff–)Tamari orders appeared in [49] (also
see [86] and references cited there). Whereas this projects, for example, B(4, 1) (permutahedron)
to T (6, 3) (associahedron), we describe a projection B(4, 1) → T (4, 1) (tetrahedron), and more
generally B(N,n)→ T (N,n).
The (N + 1)-simplex equation arises as a consistency condition of a system of N -simplex
equations. The higher Bruhat orders are also crucial for understanding this “integrability” of
the simplex equations. In the same way, the higher Tamari orders provide the combinatorial
tools to express integrability of polygon equations.
Using the transposition map P to define R := R̂P, and generalizing this to maps Rij : Vi ⊗
Vj → Vj ⊗ Vi, the Yang–Baxter equation takes the form
R23,1R13,2R12,1 = R12,2R13,1R23,2 on V1 ⊗ V2 ⊗ V3. (1.2)
Rij,a := Rij,a,a+1 acts on Vi and Vj , at positions a and a+ 1, in a product of such spaces. The
higher Bruhat orders ensure a correct matching of the two different types of indices. In fact, the
boldface indices are completely determined, they do not contain independent information. Fig. 1
shows a familiar visualization of the Yang–Baxter equation in terms of deformations of chains of
edges on a cube. Supplying the latter with the Bruhat order B(3, 0), these are maximal chains.
The information given in the caption of Fig. 1 will also be relevant for subsequent figures in this
work.
The (weak) Bruhat orders B(N, 1), N > 2, form polytopes called permutahedra. Not all
higher Bruhat orders can be realized on polytopes. The N -simplex equation is associated with
the Bruhat order B(N + 1, N − 1), but its structure is rather visible on B(N + 1, N − 2). The
latter possesses a reduction to the 1-skeleton of a polyhedron on which the simplex equation can
2See [102] for the example of the Hopf algebra of the quantum plane ab = q2ba with q a root of unity.
3This definition of higher Tamari orders emerged from our exploration of a special class of line soliton solutions
of the Kadomtsev–Petviashvili (KP) equation [24, 25].
4More precisely, HST1(n, d), as defined, e.g., in [86], is expected to be order isomorphic to T (n, d + 1), as
defined in Section 2.
4 A. Dimakis and F. Müller-Hoissen
Figure 1. The first row shows a sequence of maximal chains on half of the poset B(3, 0), which is
the Boolean lattice on {1, 2, 3}. The second row shows a corresponding sequence on the complementary
part. Glued together along their boundaries, they form a cube. The two ways of deforming the initial
lexicographically ordered maximal chain to the final, reverse lexicographically ordered chain, results in
a consistency condition, which is the Yang–Baxter equation (1.2). Edges are associated with spaces Vi, i =
1, 2, 3. For example, the second step in the first row corresponds to the action of id⊗R13 on V2⊗V1⊗V3.
The boldface indices that determine where, in a product of spaces, the map acts, correspond to the
positions of the active edges in the respective (brown) maximal chain, counting from the top downward.
Figure 2. Here the poset is T (5, 2), which also forms a cube. Edges are now numbered by 3-element
subsets of 12345 := {1, 2, 3, 4, 5}. If the edges of a face are labeled by the four 3-element subsets of
ijkl, there is a map Tijkl associated with it. This rule does not apply to the second step in the second
row, however, since here the edges of the active face involve five (rather than only four) digits. The two
possibilities of deforming the initial (lexicographically ordered) maximal chain (123, 134, 145) into the final
(reverse lexicographically ordered) maximal chain (345, 235, 125) result in the pentagon equation (1.3).
be visualized in the same way as the Yang–Baxter equation is visualized on B(3, 0) (also see [2]
for a similar view). This is elaborated in Section 3.
Also for the polygon equations, proposed in this work, all appearances of a map, like T in
the pentagon equation in Fig. 2, will be treated as a priori different maps (now on both sides of
the equation). Again we attach to them additional indices that carry combinatorial information,
now governed by higher Tamari orders.
The Tamari orders T (N, 1), N = 3, 4, . . ., T (N, 2), N = 4, 5, . . ., T (N, 3), N = 5, 6, . . ., as
defined in [25] and Section 2, form simplexes, hypercubes and associahedra (Stasheff–Tamari
polytopes), respectively. But not all Tamari orders can be realized on polytopes.
The N -gon equation is associated with T (N,N − 2), but its structure is rather revealed by
Simplex and Polygon Equations 5
T (N,N − 3). For small enough N , the latter forms a polyhedron. For higher N it admits
a polyhedral reduction. The structure of the N -gon equation can then be visualized in terms of
deformations of maximal chains on the corresponding polyhedron. Fig. 2 shows the example of
the pentagon equation, here obtained in the form
T2345,1T1245,2T1234,1 = T1235,2P1T1345,2 (1.3)
(also see Section 4). This implies (1.1) for T̂ = T P.
Section 2 first provides a brief account of higher Bruhat orders [78, 79, 80, 107]. The main
result in this section is a decomposition of higher Bruhat orders, where higher Tamari orders (in
the form introduced in [25]) naturally appear. Section 3 explains the relation between higher
Bruhat orders and simplex equations, and how the next higher simplex equation arises as a con-
sistency condition of a localized system of simplex equations. Section 4 associates in a similar
way polygon equations, which generalize the pentagon equation, with higher Tamari orders. As
in the case of simplex equations, the (N + 1)-gon equation arises as a consistency condition of
a system of localized N -gon equations. Section 5 reveals relations between simplex and polygon
equations, in particular providing a deeper explanation for and considerably generalizing a rela-
tion between the pentagon equation and the 4-simplex equation, first observed in [57]. Finally,
Section 6 contains some concluding remarks and Appendix A supplements all this by expressing
some features of simplex and polygon equations via a more abstract approach.
2 Higher Bruhat and Tamari orders
In the first subsection we recall some material about higher Bruhat orders from [78, 79, 80, 107].
The second subsection introduces a decomposition of higher Bruhat orders that includes higher
Tamari orders, in the form we defined them in [25].
2.1 Higher Bruhat orders
For N ∈ N, let [N ] denote the set {1, 2, . . . , N}. The packet P (K) of K ⊂ [N ] is the set of all
subsets of K of cardinality one less than that of K. For K = {k1, . . . , kn+1} in natural order,
i.e., k1 < · · · < kn+1, we set
−→
P (K) := (K \ {kn+1},K \ {kn}, . . . ,K \ {k1}),
←−
P (K) := (K \ {k1},K \ {k2}, . . . ,K \ {kn+1}).
The first displays the packet of K in lexicographical order (<lex). The second displays P (K) in
reverse lexicographical order.
Let
(
[N ]
n
)
, 0 ≤ n ≤ N , denote the set of all subsets of [N ] of cardinality n. Its cardinality is
c(N,n) :=
(
N
n
)
.
A linear (or total) order ρ on
(
[N ]
n
)
can be written as a sequence ρ = (J1, . . . , Jc(N,n)) with Ja ∈(
[N ]
n
)
. It is called admissible if, for all K ∈
(
[N ]
n+1
)
, ρ induces on P (K) either the lexicographical or
the reverse lexicographical order, i.e., either
−→
P (K) or
←−
P (K) is a subsequence of ρ. Let A(N,n)
denote the set of admissible linear orders of
(
[N ]
n
)
.
The envelope E(J) of J ∈
(
[N ]
n
)
is the set of K ∈
(
[N ]
n+1
)
such that J ∈ P (K). An equivalence
relation on A(N,n) is obtained by setting ρ ∼ ρ′ if ρ and ρ′ only differ by a sequence of exchanges
of neighboring elements J , J ′ with E(J) ∩ E(J ′) = ∅. We set
B(N,n) := A(N,n)/∼.
6 A. Dimakis and F. Müller-Hoissen
Example 2.1.
(
[4]
2
)
= {12, 13, 14, 23, 24, 34}, where ij := {i, j}, allows 6! = 720 linear orders,
but only 14 are admissible. For example, ρ = (12, 34, 14, 13, 24, 23) ∈ A(4, 2), since it contains
the packets of the four elements of
(
[4]
3
)
in the orders
−→
P (123),
−→
P (124),
←−
P (134),
←−
P (234). We have
ρ ∼ (34, 12, 14, 13, 24, 23) ∼ (34, 12, 14, 24, 13, 23). B(4, 2) = A(4, 2)/∼ has 8 elements.
The inversion set inv[ρ] of ρ ∈ A(N,n) is the set of all K ∈
(
[N ]
n+1
)
such that P (K) is contained
in ρ in reverse lexicographical order. All members of the equivalence class [ρ] ∈ B(N,n) have
the same inversion set. Next we introduce the inversion operation
IK :
−→
P (K) 7→
←−
P (K).
If
−→
P (K) appears in ρ ∈ A(N,n) at consecutive positions, let IKρ be the linear order obtained
by inversion of
−→
P (K) in ρ. Then5 IKρ ∈ A(N,n) and inv[IKρ] = inv[ρ]∪{K}. This corresponds
to the covering relation
[ρ]
K→ [IKρ],
which determines the higher Bruhat order on the set B(N,n) [78, 80]. In the following, we will
mostly drop the adjective “higher”. B(N,n) has a unique minimal element [α] that contains the
lexicographically ordered set
(
[N ]
n
)
, hence inv[α] = ∅, and a unique maximal element [ω] that
contains the reverse lexicographically ordered set
(
[N ]
n
)
, hence inv[ω] =
(
[N ]
n+1
)
. The Bruhat orders
are naturally extended by defining B(N, 0) as the Boolean lattice on [N ], which corresponds
to the 1-skeleton of the N -cube, with edges directed from a fixed vertex toward the opposite
vertex.
Remark 2.2. There is a natural correspondence between the elements of A(N,n+ 1) and the
maximal chains of B(N,n) [80]. Associated with σ = (K1, . . . ,Kc(N,n+1)) ∈ A(N,n + 1) is the
maximal chain
[α]
K1−→ [ρ1]
K2−→ [ρ2]
K3−→ · · ·
Kc(N,n+1)−→ [ω],
where inv[ρr] = {K1, . . . ,Kr}. This allows to construct B(N,n) from B(N,n+ 1). As a conse-
quence, all Bruhat orders B(N,n), n < N − 1, can be constructed recursively from the highest
non-trivial, which is B(N,N − 1).
Example 2.3. B(N,N−1) is simply
−→
P ([N ])
[N ]→
←−
P ([N ]). From the two admissible linear orders
−→
P ([N ]) = (N̂ , . . . , 2̂, 1̂) and
←−
P ([N ]) = (1̂, 2̂, . . . , N̂), where k̂ := [N ] \ {k} (“complementary
notation”), we can construct the two maximal chains of B(N,N − 2):
[α]
N̂−→ [ρ1]
N̂−1−→ [ρ2] −→ · · · −→ [ρN−1]
1̂−→ [ω],
[α]
1̂−→ [σ1]
2̂−→ [σ2] −→ · · · −→ [σN−1]
N̂−→ [ω]. (2.1)
The example B(4, 2) is displayed below in (2.2).
Remark 2.4. U ⊂
(
[N ]
n+1
)
is called a consistent set if, for all L ∈
(
[N ]
n+2
)
, U ∩P (L) can be ordered
in such a way that it becomes a beginning segment either of
−→
P (L) or of
←−
P (L).6 Consistent sets
are in bijective correspondence with inversion sets [107].
5Since two different packets have at most a single member in common, such an inversion does not change the
order of other packets in ρ than that of K.
6A beginning segment of a sequence is a subsequence that starts with the first member of the sequence and
contains all its members up to a final one. Also the empty sequence and the full sequence are beginning segments.
Simplex and Polygon Equations 7
23
13
12
23
13
23
13
12
12
23
12
13
23
13
12
13
23 12
23
12
13
23
13
12
12
13
12
23
12
13
13
23
23
23
13
12
12
13
12
23
13
23
Figure 3. Projection of B(4, 1) (permutahedron) to B(3, 0) (cube), each split into two complementary
parts. Here we chose k = 4 in Remark 2.5 and use complementary notation for the labels, but with hats
omitted. The coloring marks those parts in the two Bruhat orders that are related by the projection.
34
24
23
14
13
12
24
34
14
14
24
34
23
13
13
23
34
24
14
13
12
34
12
23
24
12
13
14 34
24
23
14
13
12
14
23
14
24
34
13
23
34
13
24
12
34
12
23
24
12
13
14
34
24
23
14
13
12
12
13
14
12
23
24
34
13
13
23
34
12
13
24
14
14
24
34
14
24
23
34
34
24
23
14
13
12
12
13
14
12
23
24
12
34
13
24
13
23
34
14
24
34
14
23
Figure 4. Projection of B(5, 2) (Felsner–Ziegler polyhedron [38]) to B(4, 1) (permutahedron). Here we
chose k = 5 in Remark 2.5. Again, we use complementary labeling, and the coloring marks the parts
related by the projection.
Remark 2.5. For fixed k ∈ [N + 1], we define an equivalence relation in A(N + 1, n + 1) as
follows. Let ρ
k∼ ρ′ if ρ and ρ′ only differ in the order of elements K ∈
(
[N+1]
n+1
)
with k /∈ K. For
ρ = (K1, . . . ,Kc(N+1,n+1)) ∈ A(N + 1, n+ 1), the equivalence class ρ(k) ∈ A(N + 1, n+ 1)/
k∼ is
then completely characterized by the subsequence (Ki1 , . . . ,Kic(N,n)
) consisting of only those Ki
that contain k. Hence we can identify ρ(k) with this subsequence. As a consequence, there
is an obvious bijection between A(N + 1, n + 1)/
k∼ and A(N,n). Clearly, inversions of pac-
kets P (L), L ∈
(
[N+1]
n+2
)
, with k /∈ L, have no effect on the equivalence classes. If k ∈ L,
and if
−→
P (L) appears in ρ at consecutive positions, then (ILρ)(k) is obtained from ρ(k) by the
inversion
−→
P (L) \ {L \ {k}} →
←−
P (L) \ {L \ {k}}. Since the latter naturally corresponds to
−→
P (L \ {k}) →
←−
P (L \ {k}), the bijection A(N + 1, n + 1)/
k∼→ A(N,n) is monotone, i.e.,
order-preserving. Since the equivalence relation ∼ is compatible with
k∼, the bijection induces
a corresponding monotone bijection B(N + 1, n+ 1)/
k∼ → B(N,n). We will use this projection
in Section 3.4. Examples are shown in Figs. 3 and 4.
Example 2.6. α = (123, 124, 134, 234, 125, 135, 235, 145, 245, 345) represents the minimal ele-
ment [α] of B(5, 3). Let k = 5. Then α(5) is represented by (125, 135, 235, 145, 245, 345). This
corresponds to (12, 13, 23, 14, 24, 34), which represents the minimal element of B(4, 2). Inversion
of the packet of L = {i, j,m, 5}, 1 ≤ i < j < m < 5, corresponds to inversion of the packet of
{i, j,m}, which defines an edge in B(4, 2).
2.2 Three color decomposition of higher Bruhat orders
For K ∈
(
[N ]
n+1
)
, let Po(K), respectively Pe(K), denote the half-packet of elements of P (K)
with odd, respectively even, position in the lexicographical order. We assign colors to elements
8 A. Dimakis and F. Müller-Hoissen
of
−→
P (K), respectively
←−
P (K), as follows. An element of Po(K) is blue in
−→
P (K) and red in
←−
P (K), and an element of Pe(K) is red in
−→
P (K) and blue in
←−
P (K).
Example 2.7. For K = {1, 2, 3, 4, 5} = 12345, we have Po(K) = {1234, 1245, 2345} and
Pe(K) = {1235, 1345}. Hence
−→
P (12345) = (1234, 1235, 1245, 1345, 2345),
←−
P (12345) = (2345, 1345, 1245, 1235, 1234).
We say J ∈
(
[N ]
n
)
is blue (red) in ρ ∈ A(N,n) if, for all K ∈ E(J), J is blue (red) in
−→
P (K),
respectively
←−
P (K), depending in which order P (K) appears in ρ.7 J is called green in ρ, if
there are K,K ′ ∈ E(J), such that J is blue with respect to K and red with respect to K ′.
Example 2.8. The following element of A(5, 3) has empty inversion set,
α = (123, 124, 125, 134, 135, 145, 234, 235, 245, 345).
For example, we have E(124) = {1234, 1245}, and α contains
−→
Po(1245) = (124, 145) and
−→
Pe(1234) = (124, 234) as subsequences. This shows that 124 is blue in
−→
P (1245) and red in
−→
P (1234), therefore green in α.
For each c ∈ {b, r, g} (where b, r, g stands for blue, red and green, respectively), we define an
equivalence relation on A(N,n) : ρ ∼c ρ′ if ρ and ρ′ have the same elements with color c in the
same order. Let ρ(c) denote the corresponding equivalence class, and
A(c)(N,n) := A(N,n)/∼c, c ∈ {b, r, g}.
ρ(c) can be identified with the subsequence of elements in ρ having color c.
The definition of the color of an element J of a linear order ρ only involves the inversion set
of ρ, but not ρ itself (also see footnote 7). Hence, if J has color c in ρ, then it has the same color
in any element of [ρ]. As a consequence, for each c, the equivalence relation ∼c is compatible
with ∼. Defining
B(c)(N,n) := A(c)(N,n)/∼ = (A(N,n)/∼)/∼c, c ∈ {b, r, g},
we thus obtain a projection B(N,n) → B(c)(N,n) via [ρ] 7→ [ρ(c)]. We will show that the
resulting single-colored sets inherit a partial order from the respective Bruhat order.
Lemma 2.9. Let K ∈ E(J) ∩ Po(L) (respectively, K ∈ E(J) ∩ Pe(L)) for some J ∈
(
[N ]
n
)
and
L ∈
(
[N ]
n+2
)
, where n < N − 1. Let K ′ ∈ E(J) ∩ P (L), K ′ 6= K.
If K <lex K
′, then J ∈ Po(K ′) (respectively, J ∈ Pe(K ′)).
If K ′ <lex K, then J ∈ Pe(K ′) (respectively, J ∈ Po(K ′)).
Proof. Since K,K ′ ∈ E(J) and K 6= K ′, we can write K = J ∪ {k} and K ′ = J ∪ {k′}, with
k, k′ /∈ J , k 6= k′, and L = K ∪ {k′} = K ′ ∪ {k}. K <lex K
′ is equivalent to k < k′. Let us write
−→
P (L) = (L \ {`n+2}, . . . , L \ {`1}) with `1 < `2 < · · · < `n+2.
K ∈ Po(L) (K ∈ Pe(L)) means that K = L\{k′} has an odd (even) position in
−→
P (L), hence k′
has an odd (even) position in (`n+2, . . . , `1). If k < k′, then removal of k from (`n+2, . . . , `1)
does not change this, so that k′ also has an odd (even) position in (`n+2, . . . , ǩ, . . . , `1), whereˇ
indicates an omission. It follows that J = K ′ \ {k′} ∈ Po(K
′) (J ∈ Pe(K
′)). If k′ < k,
then the position of k′ in (`n+2, . . . , ǩ, . . . , `1) is even (odd), hence J = K \ {k′} ∈ Pe(K
′)
(J ∈ Po(K ′)). �
7This means that J is blue (red) in ρ if J ∈ Po(K) (J ∈ Pe(K)) for all K ∈ E(J) \ inv[ρ], and J ∈ Pe(K)
(J ∈ Po(K)) for all K ∈ E(J) ∩ inv[ρ].
Simplex and Polygon Equations 9
In view of the bijection between A(N,n + 1) and the set of maximal chains of B(N,n), it
is natural to say that [ρ]
K→ [IKρ] has color c in some maximal chain of B(N,n) if K has this
color in the associated element of A(N,n + 1). An equivalent statement, formulated next, in
particular shows that the color of [ρ]
K→ [IKρ] is the same in any maximal chain in which it
appears, hence we can speak about [ρ]
K→ [IKρ] having color c in B(N,n).
If
−→
P (L) ∩ inv[ρ] is a beginning segment, we will say that
−→
P (L) has beginning segment with
respect to [ρ]. A corresponding formulation applies with
−→
P (L) replaced by
←−
P (L). Let n < N−1.
Then [ρ]
K→ [IKρ] is blue (red) if, for all L ∈ E(K), either
−→
P (L) has beginning segment w.r.t. [ρ]
and also w.r.t. [IKρ], and K ∈ Po(L) (K ∈ Pe(L)), or
←−
P (L) has beginning segments w.r.t. [ρ]
and [IKρ], and K ∈ Pe(L) (K ∈ Po(L)).8 Otherwise [ρ]
K→ [IKρ] is green.
Proposition 2.10. Let n < N − 1, ρ ∈ A(N,n) and K ∈
(
[N ]
n+1
)
.
(a) If [ρ]
K→ [IKρ] is blue, then the elements of Po(K) are blue in [ρ] and green in [IKρ], and
the elements of Pe(K) are blue in [IKρ] and green in [ρ].
(b) If [ρ]
K→ [IKρ] is red, then the elements of Pe(K) are red in [ρ] and green in [IKρ], and the
elements of Po(K) are red in [IKρ] and green in [ρ].
(c) If [ρ]
K→ [IKρ] is green, then all elements of P (K) are green in both, [ρ] and [IKρ].
Proof. (a) If [ρ]
K→ [IKρ] is blue, this means that for all L ∈ E(K) either (i)
−→
P (L) has beginning
segments w.r.t. [ρ] and [IKρ], and K ∈ Po(L), or (ii)
←−
P (L) has beginning segments w.r.t. [ρ]
and [IKρ], and K ∈ Pe(L). In case (i), let J ∈ P (K) and K ′ ∈ P (L) ∩ E(J). If K ′ /∈ inv[IKρ],
then K <lex K
′ and thus J ∈ Po(K ′) by Lemma 2.9. If K ′ ∈ inv[IKρ], then K ′ <lex K, hence
J ∈ Pe(K ′) according to Lemma 2.9. In both cases we can conclude that, if J ∈ Po(K), then J
is blue in [ρ] and green in [IKρ], and if J ∈ Pe(K), then J is blue in [IKρ] and green in [ρ]. The
case (ii) is treated correspondingly.
(b) is proved in the same way.
(c) [ρ]
K→ [IKρ] green means that there are L1, L2 ∈ E(K), L1 6= L2, such that one of the
following three cases holds:
(i)
−→
P (L1) and
−→
P (L2) have beginning segments w.r.t. [ρ] and [IKρ], and K ∈ Po(L1)∩Pe(L2),
(ii)
−→
P (L1) and
←−
P (L2) have beginning segments w.r.t. [ρ] and [IKρ], and K ∈ Po(L1)∩Po(L2)
or K ∈ Pe(L1) ∩ Pe(L2),
(iii)
←−
P (L1) and
←−
P (L2) have beginning segments w.r.t. [ρ] and [IKρ], and K ∈ Pe(L1)∩Po(L2).
In case (i), let J ∈ P (K) and K1 ∈ P (L1) ∩ E(J), K2 ∈ P (L2) ∩ E(J), K1 6= K2. If
K <lex K1,K2, then K1,K2 /∈ inv[IKρ], hence J ∈ Po(K1) ∩ Pe(K2) by Lemma 2.9. If K1 <lex
K <lex K2, then K1 ∈ inv[IKρ] and K2 /∈ inv[IKρ], so that J ∈ Pe(K1)∩Pe(K2) by Lemma 2.9.
If K2 <lex K <lex K1, then K1 /∈ inv[IKρ] and K2 ∈ inv[IKρ], hence J ∈ Po(K1) ∩ Po(K2).
Finally, if K1,K2 <lex K, then K1,K2 ∈ inv[IKρ], hence J ∈ Pe(K1) ∩ Po(K2) by Lemma 2.9.
In all these cases J is green in both, [ρ] and [IKρ]. The cases (ii) and (iii) can be treated in
a similar way. �
The preceding proposition in particular shows that blue (red) elements of [ρ] are not affected
by red (blue) and green inversions.
8Both conditions covered by “has beginning segments with respect to [ρ] and [IKρ]” are necessary in order to
avoid ambiguities that would otherwise arise if P (L)∩ inv[ρ] is empty or if P (L)∩ inv[IKρ] is the full packet. We
are grateful to one of the referees for pointing this out.
10 A. Dimakis and F. Müller-Hoissen
Proposition 2.11. Let [ρ]
K→ [IKρ] be blue (red) in B(N,n), n < N . Then any J ∈ P (K) that
is blue (red) in [ρ] is not blue (red) in any subsequent element of B(N,n).
Proof. This is obvious if n = N − 1. Let n < N − 1, [ρ]
K→ [IKρ] and J ∈ P (K) blue.
According to Proposition 2.10, J is green in [ρ1] = [IKρ]. Let us assume that J becomes
blue again in some subsequent [ρr]. Then [ρr−1]
Kr→ [ρr] has to be blue and J ∈ Pe(Kr). As
a consequence, K,Kr ∈ E(J), K 6= Kr, and L = K ∪ Kr has cardinality n + 2. If
−→
P (L) has
beginning segment w.r.t. [ρr−1], then also w.r.t. [ρ1]. Since K ∈ Po(L) (because [ρ]
K→ [ρ1] is
blue) and K <lex Kr, Lemma 2.9 yields J ∈ Po(Kr) and thus a contradiction. If
←−
P (L) has
beginning segment w.r.t. [ρr−1], then K ∈ Pe(L) and Kr <lex K, hence J ∈ Po(Kr) according
to Lemma 2.9, so we have a contradiction. The red case is treated in the same way. �
Proposition 2.12. Let [α] and [ω] be the minimal and the maximal element of B(N,n), re-
spectively. Then the blue (red) elements of [α] are the red (blue) elements of [ω]. Furthermore,
[α] and [ω] share the same green elements.
Proof. Let J be blue in [α]. Since inv[α] = ∅, J is blue in
−→
P (K) for all K ∈ E(J). Since
inv[ω] contains all K, J is red in [ω]. Correspondingly, a red J in [α] is blue in [ω]. The last
statement of the proposition is then obvious. �
Let us recall that we also use ρ(c) to denote the subsequence of ρ ∈ A(N,n) of color c.
Now any ρ ∈ A(N,n) can be decomposed into three subsequences, ρ(b), ρ(r) and ρ(g), and this
decomposition is carried over to B(N,n).
For K ∈
(
[N ]
n+1
)
, let us introduce the half-packet inversions
I
(b)
K :
−→
Po(K)→
←−
Pe(K), I
(r)
K :
−→
Pe(K)→
←−
Po(K).
Let ρ
K→ IKρ be an inversion in A(N,n). From the above propositions we conclude:
• if the inversion is blue, then (IKρ)(b) = I
(b)
K ρ(b), (IKρ)(r) = ρ(r), and (IKρ)(g) = I
(r)
K ρ(g),
• if the inversion is red, then (IKρ)(b) = ρ(b), (IKρ)(r) = I
(r)
K ρ(r), and (IKρ)(g) = I
(b)
K ρ(g),
• if the inversion is green, then (IKρ)(b) = ρ(b), (IKρ)(r) = ρ(r), and (IKρ)(g) = IKρ
(g).
In the following, B(c)(N,n) shall denote the corresponding set supplied with the induced partial
order. B(b)(N,n) is the (higher) Tamari order T (N,n) (see [25] for an equivalent definition).
B(r)(N,n) is the dual of the (higher) Tamari order T (N,n). B(g)(N,n) will be called mixed
order. The latter involves all the three inversions. It should be noted that a red (blue) half-
packet inversion in B(g)(N,n) stems from a blue (red) inversion IK .
Remark 2.13. B(b)(N,N − 1) is
−→
Po([N ])
[N ]→
←−
Pe([N ]) and B(r)(N,N − 1) is
−→
Pe([N ])
[N ]→
←−
Po([N ]).
B(g)(N,N − 1) is empty. B(N,N − 2) consists of a pair of maximal chains, see (2.1). We set
m := N mod 2. The two blue subchains
[α(b)]
N̂−→ [ρ
(b)
1 ]
N̂−2−→ [ρ
(b)
3 ] −→ · · · −→ [ρ
(b)
N+m−5]
4̂−m−→ [ρ
(b)
N+m−3]
2̂−m−→ [ω(b)],
[α(b)]
m̂+1−→ [σ
(b)
m+1]
m̂+3−→ [σ
(b)
m+3] −→ · · · −→ [σ
(b)
N−5]
N̂−3−→ [σ
(b)
N−3]
N̂−1−→ [ω(b)],
constitute B(b)(N,N−2). All inversions are blue, of course. Correspondingly, the two red chains
[α(r)]
N̂−1−→ [ρ
(r)
2 ]
N̂−3−→ [ρ
(r)
4 ] −→ · · · −→ [ρ
(r)
N−m−4]
m̂+3−→ [ρ
(r)
N−m−2]
m̂+1−→ [ω(r)],
Simplex and Polygon Equations 11
Figure 5. Structure of the chains (2.2). A rhombus with two green and two blue (red) edges corresponds
to a blue (red) inversion in B(4, 2). Such a rhombus corresponds to a hexagon in B(4, 1) (see Remark 5.1).
[α(r)]
2̂−m−→ [σ
(r)
2−m]
4̂−m−→ [σ
(r)
4−m] −→ · · · −→ [σ
(r)
N−4]
N̂−2−→ [σ
(r)
N−2]
N̂−→ [ω(r)],
form B(r)(N,N − 2). All inversions are red. One maximal chain of B(g)(N,N − 2) is
[α(g)]
N̂−→ [ρ
(g)
1 ]
N̂−1−→ [ρ
(g)
2 ] −→ · · · −→ [ρ
(g)
N−2]
2̂−→ [ρ
(g)
N−1]
1̂−→ [ω(g)],
where now an inversion
k̂→ is blue if k̂ ∈ Pe([N ]) and red if k̂ ∈ Po([N ]). The second chain is
[α(g)]
1̂−→ [σ
(g)
1 ]
2̂−→ [σ
(g)
2 ] −→ · · · −→ [σ
(g)
N−2]
N̂−1−→ [σ
(g)
N−1]
N̂−→ [ω(b)],
where
k̂→ is blue if k̂ ∈ Po([N ]) and red if k̂ ∈ Pe([N ]). There are no green inversions in this
case.
Example 2.14. B(4, 2) consists of the two maximal chains
12
13
14
23
24
34
∼→
12
13
23
14
24
34
123→
23
13
12
14
24
34
124→
23
13
24
14
12
34
∼→
23
24
13
14
34
12
134→
23
24
34
14
13
12
234→
34
24
23
14
13
12
12
13
14
23
24
34
234→
12
13
14
34
24
23
134→
12
34
14
13
24
23
∼→
34
12
14
24
13
23
124→
34
24
14
12
13
23
123→
34
24
14
23
13
12
∼→
34
24
23
14
13
12
(2.2)
Here they are resolved into admissible linear orders, i.e., elements of A(4, 2). These are the
maximal chains of B(4, 1) (forming a permutahedron). The blue and red subchains of (2.2),
forming B(b)(4, 2) and B(r)(4, 2), respectively, are
12
23
34
123→ 13
34
134→ 14
12
23
34
234→ 12
24
124→ 14 14
124→ 24
12
234→
34
23
12
14
134→ 34
13
123→
34
23
12
B(g)(4, 2) is given by
13
24
123→
23
12
24
124→ 23
14
134→
23
34
13
234→ 24
13
13
24
234→
13
34
23
134→ 14
23
124→
24
12
23
123→ 24
13
Fig. 5 displays the structure of the two maximal chains (2.2) of B(4, 2).9
9In the context of Soergel bimodules, a corresponding diagrammatic equation (also see Fig. 27) appeared in [31]
as the A3 Zamolodchikov relation.
12 A. Dimakis and F. Müller-Hoissen
Example 2.15. In the following, we display one of the maximal chains of B(5, 2), resolved into
linear orders (elements of A(5, 2)), and its single-colored subsequences
12
13
14
15
23
24
25
34
35
45
∼→
12
13
23
14
15
24
25
34
35
45
123→
23
13
12
14
15
24
25
34
35
45
∼→
23
13
12
14
24
15
25
34
35
45
124→
23
13
24
14
12
15
25
34
35
45
125→
23
13
24
14
25
15
12
34
35
45
∼→
23
24
13
14
34
25
15
12
35
45
134→
23
24
34
14
13
25
15
12
35
45
∼→
23
24
34
14
25
13
15
35
12
45
135→
23
24
34
14
25
35
15
13
12
45
∼→
23
24
34
25
35
14
15
45
13
12
145→
23
24
34
25
35
45
15
14
13
12
234→
34
24
23
25
35
45
15
14
13
12
235→
34
24
35
25
23
45
15
14
13
12
∼→
34
35
24
25
45
23
15
14
13
12
245→
34
35
45
25
24
23
15
14
13
12
345→
45
35
34
25
24
23
15
14
13
12
The three subsequences collapse to
12
23
34
45
123→
13
34
45
134→ 14
45
145→ 15 15
125→ 25
12
235→
35
23
12
345→
45
34
23
12
13
14
24
25
35
123→
23
12
14
24
25
35
124→
23
24
14
12
25
35
125→
23
24
14
15
35
134→
23
24
34
13
15
35
135→
23
24
34
35
15
13
145→
23
24
34
35
45
14
13
234→
34
24
23
35
45
14
13
235→
34
24
25
45
14
13
245→
34
45
25
24
14
13
345→
35
25
24
14
13
Here the blue order is ruled by I
(b)
ijk : (ij, jk) 7→ (ik), the red order by I
(r)
ijk : (ik) 7→ (jk, ij), where
1 ≤ i < j < k ≤ 5. The green order involves these two and in addition Iijk : (ij, ik, jk) 7→
(jk, ik, ij).
Remark 2.16. Inherited from the Bruhat orders, for c ∈ {b, r}, there is a one-to-one correspon-
dence between elements of A(c)(N,n + 1) and maximal chains of B(c)(N,n). No such relation
exists for the mixed order, but elements of A(g)(N,n+ 1) are in one-to-one correspondence with
the green inversion subsequences of maximal chains of B(g)(N,n).
Remark 2.17. In Remark 2.5 we defined, for each k ∈ [N + 1], a projection B(N + 1, n+ 1)→
B(N,n), via an equivalence relation
k∼. If k ∈ [N + 1] \ {1, N + 1}, these projections do not
respect the above three color decomposition. The reason is that if L ∈
(
[N+1]
n+2
)
contains k, then
Po(L) \ {L \ {k}} and Pe(L) \ {L \ {k}} cannot be brought into natural correspondence with
the half-packets Po(L \ {k}) and Pe(L \ {k}), respectively. For example, if L = 1234 and k = 2,
then L \ {k} = 134, Po(L) = {123, 134}, Pe(L) = {124, 234}, hence Po(L) \ {L \ {k}} = {123},
Pe(L) \ {L \ {k}} = {124, 234}, while Po(L \ {k}) = {13, 34} and Pe(L \ {k}) = {14}. But if
k = 1, then L \ {1} is the last element of
−→
P (L) and its elimination thus does not influence
the positions of the remaining elements. For k = 1 we therefore obtain monotone projections
B(c)(N +1, n+1)→ B(c)(N,n). The other exception is k = N +1. Then L\{N +1} is the first
element of
−→
P (L) and its elimination turns odd into even elements, and vice versa. In this case
we obtain monotone projections B(b)(N+1, n+1)→ B(r)(N,n), B(r)(N+1, n+1)→ B(b)(N,n)
and B(g)(N + 1, n+ 1)→ B(g)(N,n). We will use the projection with k = 1 in Section 4.4. See,
in particular, Figs. 23 and 24.
Simplex and Polygon Equations 13
3 Simplex equations
In this section we consider realizations of Bruhat orders in terms of sets and maps between Carte-
sian10 products of the sets. The N -simplex equation is directly associated with B(N +1, N−1),
but its structure is fully displayed as a polyhedral reduction of B(N + 1, N − 2). Section 3.1
explains the relation with polyhedra and prepares the stage for the definition of simplex equa-
tions in Section 3.2, which contains explicit expressions up to the 7-simplex equation, and the
associated polyhedra. Section 3.3 discusses the integrability of simplex equations. The reduction
of the Bruhat order B(N + 2, N) to B(N + 1, N − 1) induces a reduction of the (N + 1)-simplex
equation to the N -simplex equation. This is the subject of Section 3.4.
3.1 Resolutions of B(N + 1, N − 1) and polyhedra
Let sa denote the operation of exchange of elements Ja, Ja+1 of ρ = (J1, . . . , Jc(N+1,n)) ∈ A(N +
1, n), which is applicable (only) if E(Ja) ∩E(Ja+1) = ∅. For any β, β′ ∈ [ρ], there is a minimal
number m of exchange operations sa1 , . . . , sam , such that β′ = sβ′,ββ, where sβ′,β := sam · · · sa1 .
The sequence β0, β1, . . . , βm, where β0 = β, βi = saiβi−1, i = 1, . . . ,m, is called a resolution
of [ρ] from β to β′. It is unique up to potential applications of the identities
sasb = sbsa if |a− b| > 1, sasa+1sa = sa+1sasa+1. (3.1)
Let C : [ρ0]
K1−→ [ρ1]
K2−→ · · · Kk−→ [ρk] be a chain in B(N + 1, n). A resolution C̃ of C is a sequence
of resolutions of all [ρi], such that the initial element of the resolution of [ρi+1] is obtained by
application of IKi+1 to the final element of the resolution of [ρi], for i = 0, . . . , k − 1.
Denoting by ιa an inversion, acting at positions a, a + 1, . . . , a + n, of some element of
A(N + 1, n), the resolution C̃ uniquely corresponds to a composition of exchange and inver-
sion operations,
OC̃ := sβ′k,βkιak · · · sβ′1,β1
ιa1sβ′0,β0
, (3.2)
where βi is the initial and β′i the final element of the resolution of [ρi], and βi+1 = ιai+1β
′
i.
Remark 3.1. The operations sa and ιb satisfy the following identities,
saιb = ιbsa if a < b− 1 or a > b+ n,
ιaιb = ιbιa if |b− a| > n,
ιasa+n · · · sa+1sa = sa+n · · · sa+1saιa+1. (3.3)
In the last identity, sa+n · · · sa+1sa exchanges the element at position a with the block of elements
at positions a+1, . . . , a+n+1. The identities (3.3) take care of the fact that the above definition
of a resolution of a chain in B(N + 1, n) does not in general fix all the final elements of the
resolutions of the [ρi]. Using s2
a = id, in the last of the above relations one can move exchange
operations from one side to the other. Since the relations (3.1) and (3.3) are homogeneous, all
resolutions with the same initial and the same final element have the same length.
The Bruhat order B(N + 1, N − 1) consists of the two maximal chains11
Clex : [α]
[N+1]\{N+1}
−−−−−−−→ [ρ1]
[N+1]\{N}
−−−−−−−→ · · ·
[N+1]\{2}
−−−−−−−→ [ρN ]
[N+1]\{1}
−−−−−−−→ [ω],
10Alternatively, we may as well consider tensor products or direct sums, assuming that the sets carry the
necessary additional structure.
11Here “lex” and “rev” stand for “lexicographically ordered” and “reverse lexicographically ordered”, respec-
tively. In these chains we should better use complementary notation, l̂ := [N + 1]\{l}, and we will do this mostly
in the following.
14 A. Dimakis and F. Müller-Hoissen
Figure 6. Complementary sides of B(6, 3). Because of the “small cubes” (here marked purple), B(6, 3)
is not polyhedral.
Crev : [α]
[N+1]\{1}
−−−−−−−→ [σ1]
[N+1]\{2}
−−−−−−−→ · · ·
[N+1]\{N}
−−−−−−−→ [σN ]
[N+1]\{N+1}
−−−−−−−→ [ω].
This implies that there are resolutions C̃lex and C̃rev of Clex and Crev, respectively, both starting
with α and both ending with ω,
α ω
C̃lex
C̃rev
Via the correspondence between elements of A(N + 1, N − 1) and maximal chains of B(N +
1, N − 2) (see Remark 2.2), each of the two resolutions corresponds to a sequence of maximal
chains of B(N + 1, N − 2). For ρ ∈ A(N + 1, N − 1), let Cρ be the corresponding maximal chain
of B(N + 1, N − 2). C̃lex, respectively C̃rev, is then a rule for deforming Cα stepwise into Cω.
Moreover, a resolution of B(N + 1, N −1), represented by the above diagram, contains a rule
to construct a polyhedron. Starting from a common vertex, we represent the elements of α and
ω = rev(α) (α in reversed order) from top to bottom as the edges of the left, respectively right
side of a regular N(N + 1)-gon. Then we deform the left side (corresponding to α) stepwise,
following the resolution C̃lex and ending in the right side (corresponding to ω) of the polygon.
For any appearance of an exchange operation s we insert a rhombus, and for any inversion ι
a 2N -gon. This is done in such a way that opposite edges are parallel and have equal length, so
the inserted polygons are zonogons. We proceed in the same way with the resolution C̃rev. The
resulting two zonotiles constitute complementary sides of a zonohedron. Up to “small cubes”
(see the following remark), it represents B(N + 1, N − 2).
Remark 3.2. For the first few values of N , B(N + 1, N − 2) forms a polyhedron. This is
no longer so for higher values, because “small cubes” appear [38]. Fig. 6 displays them for
B(6, 3). A small cube is present in the Bruhat order B(N + 1, N − 2) whenever there are two
different resolutions of an element of B(N + 1, N − 1), which are identical except that one
of them contains a subsequence β, saβ, sa+1saβ, β
′ and the other β, sa+1β, sasa+1β, β
′ instead,
where β′ = sasa+1saβ ≡ sa+1sasa+1β. The six members of the two subsequences determine six
maximal chains of B(N + 1, N − 2), which enclose a cube (similarly as in Fig. 1). The process
of deformations of maximal chains described above keeps only half of any small cube. The
polyhedron, constructed in the way described above, is a polyhedral reduction of B(N+1, N−2).
Simplex and Polygon Equations 15
3.2 Simplex equations and associated polyhedra
With each J ∈
(
[N+1]
n
)
, we associate a set UJ . With ρ ∈ A(N + 1, n), ρ = (J1, . . . , Jc(N+1,n)), we
then associate the Cartesian product
Uρ := UJ1 × UJ2 × · · · × UJc(N+1,n)
.
Furthermore, for each K ∈
(
[N+1]
n+1
)
, let there be a map
RK : U−→
P (K)
→ U←−
P (K)
.
If U−→
P (K)
appears in Uρ at consecutive positions, starting at position a, we extend RK to a map
RK,a : Uρ → Uρ′ , where it acts non-trivially only on the sets labeled by the elements of P (K).
RK,a then represents the inversion operation ιa. The exchange operation sa will be represented
by the transposition map Pa (where P : (u, v) 7→ (v, u)), which acts at positions a and a + 1
of Uρ.12 In this way, the resolution C̃ of the chain C considered in Section 3.1 translates, via (3.2),
to a composition of maps,
RC̃ := Pβ′k,βkRKk,ak · · · Pβ′1,β1
RK1,a1Pβ′0,β0
,
where Pβ′i,βi = Pai,mi · · · Pai,1 .
Let us now turn to B(N + 1, N −1). Choosing α as the lexicographically ordered set
([N+1]
N−1
)
,
and ω as α in reverse order, we define the N -simplex equation as
RC̃lex
= RC̃rev
, (3.4)
where C̃lex and C̃rev are resolutions of Clex and Crev, respectively, with initial element α and final
element ω, and RC̃lex
, RC̃rev
are the corresponding compositions of maps RK,a, Pb. (3.4) is
independent of the choices of the resolutions C̃lex and C̃rev, since Pa and RK,b clearly satisfy
all the relations that sa and ιb fulfill (see Section 3.1). Since we have the freedom to move
a transposition in leftmost or rightmost position from one side of (3.4) to the other, the above
choice of α is no restriction.
Remark 3.3. Let
PK : U←−
P (K)
→ U−→
P (K)
be a composition of transposition maps Pa corresponding to a reversion. The maps R̂K that
we will encounter in this section are related to the respective maps RK via
R̂K = RKPK ,
and they are endomorphisms
R̂K : U←−
P (K)
−→ U←−
P (K)
.
RK acts on Uρ only if
−→
P (K) appears at consecutive positions in ρ, and it changes the order of
the factors of Uρ. In contrast, R̂K only acts on Uω (not necessarily at consecutive positions). It
does not change the order of U ’s.
12We use boldface “position” numbers in order to distinguish them more clearly from the numbers specifying
some K ∈
(
[N+1]
n+1
)
.
16 A. Dimakis and F. Müller-Hoissen
In complementary notation, the reverse lexicographical order ω on
([N+1]
N−1
)
reads
ω =
(
1̂2, 1̂3, . . . , ̂1(N + 1), 2̂3, 2̂4, . . . , ̂2(N + 1), . . . , ̂N(N + 1)
)
.
The N -simplex equation has the form
R̂1̂,A1
R̂2̂,A2
· · · R̂
N̂+1,AN+1
= R̂
N̂+1,AN+1
R̂N̂,AN
· · · R̂1̂,A1
,
where both sides are maps Uω → Uω. One has to determine the positions, given by the multi-
index Ak, of the factors of Uω, on which the map R̂k̂ acts. For the examples in this section, it
is given by Ak = (ak,1, . . . ,ak,N+1), where the integers ak,j are determined by
ak,j =
{
1
2(2n− k)(k − 1) + j if k ≤ j,
aj,k−1 if k > j.
1-simplex equation. In case of B(2, 0) we consider maps R1,R2 : U∅ −→ U∅ subject to
R2R1 = R1R2,
which is the 1-simplex equation.
2-simplex equation and the cube. Associated with the two maximal chains of B(3, 1) is
the 2-simplex, or Yang–Baxter equation,
R23,1R13,2R12,1 = R12,2R13,1R23,2,
for maps Rij : Ui ×Uj → Uj ×Ui, i < j. The two sides of this equation correspond to sequences
of maximal chains on two complementary sides of the cube, formed by B(3, 0), see Fig. 1. In
complementary notation, 23 = 1̂, 13 = 2̂ and 12 = 3̂, the Yang–Baxter equation reads
R1̂,1R2̂,2R3̂,1 = R3̂,2R2̂,1R1̂,2.
In terms of R̂k̂ := Rk̂P, it takes the form
R̂1̂,12R̂2̂,13R̂3̂,23 = R̂3̂,23R̂2̂,13R̂1̂,12.
3-simplex equation and the permutahedron. The two maximal chains of B(4, 2) are
Clex : [α]
123→ [ρ1]
124→ [ρ2]
134→ [ρ3]
234→ [ω], Crev : [α]
234→ [σ1]
134→ [σ2]
124→ [σ3]
123→ [ω].
Let us start with the lexicographical linear order α = (12, 13, 14, 23, 24, 34). The minimal element
[α] ∈ B(4, 2) also contains (12, 13, 23, 14, 24, 34). ω is α in reverse order. We already dis-
played C̃lex and C̃rev in (2.2). From them we read off
RC̃lex
= R234,1R134,3P5P2R124,3R123,1P3, RC̃rev
= P3R123,4R124,2P4P1R134,2R234,4,
for maps Rijk : Uij × Uik × Ujk → Ujk × Uik × Uij , i < j < k. This determines the 3-simplex
equation
R234,1R134,3P5P2R124,3R123,1P3 = P3R123,4R124,2P4P1R134,2R234,4.
In complementary notation, 234 = 1̂, 134 = 2̂, etc., it reads
R1̂,1R2̂,3P5P2R3̂,3R4̂,1P3 = P3R4̂,4R3̂,2P4P1R2̂,2R1̂,4, (3.5)
Simplex and Polygon Equations 17
Figure 7. The left-hand side of the 3-simplex equation (3.5) corresponds to a sequence of maximal
chains of B(4, 1). This sequence forms one side of the permutahedron in three dimensions. Here and in
the following figures, if not stated otherwise, edge labels in graphs will be in complementary notation,
but with hats omitted.
Figure 8. The right-hand side of the 3-simplex equation (3.5) corresponds to a sequence of maximal
chains of B(4, 1), forming the side of the permutahedron complementary to that in Fig. 7.
where, for example, R1̂ : U1̂4 × U1̂3 × U1̂2 → U1̂2 × U1̂3 × U1̂4 and R2̂ : U2̂4 × U2̂3 × U1̂2 →
U1̂2 × U2̂3 × U2̂4. Left- and right-hand side of (3.5) correspond, respectively, to Figs. 7 and 8.
Collapsing the sequences of graphs in these figures, we can represent the equation as in Fig. 9.
Disregarding the indices associated with the underlying Bruhat order, this is Fig. 17 in [70]
and Fig. 5 in Chapter 6 of [21], where the 3-simplex equation has been called “permutohedron
equation”.
In terms of R̂ := RP13, where Pab is the transposition map acting at positions a and b, the
3-simplex equation takes the form
R̂1̂,123R̂2̂,145R̂3̂,246R̂4̂,356 = R̂4̂,356R̂3̂,246R̂2̂,145R̂1̂,123,
which is also known as the tetrahedron or Zamolodchikov equation. Ignoring the boldface in-
dices and interpreting the others as “position indices”, we formally obtain the Frenkel–Moore
version [39]
R̂234R̂134R̂124R̂123 = R̂123R̂124R̂134R̂234.
With a different interpretation of the indices, this equation appeared, for example, in [20].
4-simplex equation and the Felsner–Ziegler polyhedron. In case of B(5, 3) we consider
maps Rijkl : Uijk × Uijl × Uikl × Ujkl → Ujkl × Uikl × Uijl × Uijk, i < j < k < l. Turning
to complementary notation, we have, for example, R2345 = R1̂ : U1̂5 × U1̂4 × U1̂3 × U1̂2 →
18 A. Dimakis and F. Müller-Hoissen
34
24
23
14
13
12
14
23
14
24
34
13
23
34
13
24
12
34
12
23
24
12
13
14
34
24
23
14
13
12
12
13
14
12
23
24
12
34
13
24
13
23
34
14
24
34
14
23
=
Figure 9. Graphical representation of the 3-simplex equation.
45
35
34
25
24
23
15
14
13
12
25
24
15
34
15
35
45
14
14
24
34
45
35
23
13
13
23
34
35
24
14
13
25
45
35
12
34
12
23
24
25
12
13
14
15
45
35
34
25
24
23
15
14
13
12
12
13
14
15
12
23
24
25
34
35
13
13
23
34
35
12
13
45
24
25
14
35
14
24
34
45
15
25
35
45
14
24
23
25
34
=
15
Figure 10. The two sides of the 4-simplex equation correspond to sequences of maximal chains on two
complementary sides (left and right figure) of the Felsner–Ziegler polyhedron, which carries the partial
order B(5, 2). Edge labels are in complementary notation, but with hats omitted.
U1̂2 × U1̂3 × U1̂4 × U1̂5. The maps are subject to the 4-simplex equation
R1̂,1R2̂,4P7P8P9P3P2P4R3̂,5P8P7P4P3R4̂,4P7R5̂,1P4P5P6P3
= P7P4P5P6P3R5̂,7R4̂,4P7P8P6P3P2P1R3̂,3P6P7P2P3R2̂,4R1̂,7. (3.6)
This can be read off from B(5, 2), which forms the Felsner–Ziegler polyhedron (G5 in [38]), see
Fig. 10. In terms of R̂k̂ := Rk̂ P23 P14, the 4-simplex equation takes the more concise form
R̂1̂,1,2,3,4R̂2̂,1,5,6,7R̂3̂,2,5,8,9R̂4̂,3,6,8,10R̂5̂,4,7,9,10
= R̂5̂,4,7,9,10R̂4̂,3,6,8,10R̂3̂,2,5,8,9R̂2̂,1,5,6,7R̂1̂,1,2,3,4. (3.7)
It is also known as the Bazhanov–Stroganov equation (see, e.g., [76]).
5-simplex equation. Turning to B(6, 4), we are dealing with maps
Rijklm : Uijkl × Uijkm × Uijlm × Uiklm × Ujklm −→ Ujklm × Uiklm × Uijlm × Uijkm × Uijkl,
where i < j < k < l < m. In complementary notation, for example,
R23456 = R1̂ : U1̂6 × U1̂5 × U1̂4 × U1̂3 × U1̂2 −→ U1̂2 × U1̂3 × U1̂4 × U1̂5 × U1̂6.
Simplex and Polygon Equations 19
56
46
45
36
35
34
26
25
24
23
16
15
14
13
12
36
26
35
16
45
16
26
36
46
56
25
15
15
25
35
45
56
46
34
24
14 14
14
24
34
45
46
35
36
23
13
13
23
34
35
36
24
25
15
14
13
26
56
46
45
36
35
12
34
12
23
24
25
26
12
13
14
15
16
56
46
45
36
35
34
26
25
24
23
16
15
14
13
12
12
13
14
15
16
12
23
24
25
26
34
35
36
13
13
23
34
35
36
45
46
24
14
14
24
34
45
46
13
14
56
35
25
26
36
15
46
15
25
35
45
56
16
26
46
56
14
15
24
16
23
25
35
26
34
36
45
36
12
=
Figure 11. The two sides of the 5-simplex equation correspond to sequences of maximal chains on two
complementary sides (left and right figure) of a polyhedral reduction of B(6, 3). Parallel edges carry the
same label.
These maps have to satisfy the 5-simplex equation
R1̂,1R2̂,5P9P10P11P12P13P14P4P3P2P5P4P6R3̂,7P11P12P13P10
P11P12P6P5P4P3P7P6R4̂,7P11P12P10P9P6P5P4R5̂,5P9P10P11P8
R6̂,1P5P6P7P8P9P10P4P5P6P3
= P12P9P10P11P8P5P6P7P8P9P10P4P5P6P3R6̂,11R5̂,7P11P12P13
P10P11P9P6P5P4P3P2P1R4̂,5P9P10P11P12P8P9P4P3P2P5P4P3
R3̂,5P9P10P11P4P5P6R2̂,7R1̂,11, (3.8)
see Fig. 11. The Bruhat order B(6, 3) is not polyhedral [38], because of the existence of “small
cubes”, see Remark 3.2 and Fig. 6. We can rewrite the 5-simplex equation as follows to display
the respective appearances (in brackets),
R1̂,1R2̂,5P9P10(P12P11P12)P13P14P4P3P2P5P6R3̂,7P11P12P13P10P11
P6(P4P5P4)P3P7P6R4̂,7(P12P11P12)P10P9P6P5P4R5̂,5P9P10P11R6̂,1
P5P6(P8P7P8)P9P10P4P5P6
= P9P10P11P5P6(P8P7P8)P9P10P4P5P6R6̂,11R5̂,7P11P12P13P10
P9P6P5(P3P4P3)P2P1R4̂,5P9(P11P10P11)P12P8P9P4P3P2P5P4
R3̂,5P9P10P11(P3P4P3)P5P6R2̂,7R1̂,11.
In terms of R̂ := RP24P15, the 5-simplex equation takes the form
R̂1̂,1,2,3,4,5R̂2̂,1,6,7,8,9R̂3̂,2,6,10,11,12R̂4̂,3,7,10,13,14R̂5̂,4,8,11,13,15R̂6̂,5,9,12,14,15
= R̂6̂,5,9,12,14,15R̂5̂,4,8,11,13,15R̂4̂,3,7,10,13,14R̂3̂,2,6,10,11,12R̂2̂,1,6,7,8,9R̂1̂,1,2,3,4,5.
6-simplex equation. In case of B(7, 5) we consider maps
Rijklmn : Uijklm × Uijkln × Uijkmn × Uijlmn × Uilmn × Ujklmn
−→ Ujklmn × Uiklmn × Uijlmn × Uijkmn × Uijkln × Uijklm,
20 A. Dimakis and F. Müller-Hoissen
6757
56
47
46
45
37
36
35
34
27
26
25
24
23
17
16
15
14
13
12
37
27
46
17
56
17
27
37
47
5767
36
26
16
16
26
36
46
56
67
57
45
35
25
15
15
25
35
45
56
57
46
47
34
24
14
14
24
34
45
46
47
35
36
37
23
13
13
23
34
35
36
37
24
25
26
16
15
14
13
27
67 57
56
47
46
45
37
36
35
12
34
12
23
24
25
26
27
12 13
14
15
16
17
6757
56
47
46
45
37
36
35
34
27
26
25
24
23
17
16
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14
13 12
12 13
14
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35
36
37
13
13
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34
35
36
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47
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14
14
24
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46
47
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57
35
25
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15
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57
12 13
67
46
36
26
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37
47
16
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67
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5767
14
15
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17
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35
27
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56
=
Figure 12. The two sides of the 6-simplex equation correspond to sequences of maximal chains on two
complementary sides (left and right figure) of a polyhedral reduction of B(7, 4).
where i < j < k < l < m < n. Turning to complementary notation, these maps have to satisfy
the 6-simplex equation
R1̂,1R2̂,6(P11P12P13P14P15P16P17P18P19P20)(P5P4P3P2)(P6P5P4)(P7P6)
P8R3̂,9(P14P15P16P17P18P19)(P13P14P15P16P17P18)(P8P7P6P5P4P3)
(P9P8P7P6)(P10P9)R4̂,10(P15P16P17P18)(P14P15P16)(P13P14P15)
(P9P8P7P6P5P4)(P10P9P8)R5̂,9(P14P15P16P17)(P13P14)(P12P11)
(P8P7P6P5)R6̂,6(P11P12P13P14P15P16)(P10P11P12)P9R7̂,1
(P6P7P8P9P10P11P12P13P14P15)(P5P6P7P8P9P10)(P4P5P6)P3
= P18(P15P16P17)P14(P11P12P13P14P15P16)(P10P11P12)P9
(P6P7P8P9P10P11P12P13P14P15)(P5P6P7P8P9P10)(P4P5P6)P3R7̂,16R6̂,11
(P16P17P18P19)(P15P16P17)(P14P15)P13(P10P9P8P7P6P5P4P3P2P1)R5̂,8
(P13P14P15P16P17P18)(P12P13P14P15)(P11P12)(P7P6P5P4P3P2)
(P8P7P6P5P4P3)R4̂,7(P12P13P14P15P16P17)(P11P12P13)(P6P5P4)(P7P6P5)
(P8P7P6)R3̂,8(P13P14P15P16)P7P8P9P10R2̂,11R1̂,16.
The two sides of this equation correspond to sequences of maximal chains on complementary
sides of a polyhedron, see Fig. 12. In terms of R̂ = RP34P25P16, the 6-simplex equation takes
the form
R̂1̂,1,2,3,4,5,6R̂2̂,1,7,8,9,10,11R̂3̂,2,7,12,13,14,15R̂4̂,3,8,12,16,17,18R̂5̂,4,9,13,16,19,20
R̂6̂,5,10,14,17,19,21R̂7̂,6,11,15,18,20,21
= R̂7̂,6,11,15,18,20,21R̂6̂,5,10,14,17,19,21R̂5̂,4,9,13,16,19,20R̂4̂,3,8,12,16,17,18
R̂3̂,2,7,12,13,14,15R̂2̂,1,7,8,9,10,11R̂1̂,1,2,3,4,5,6. (3.9)
Simplex and Polygon Equations 21
7-simplex equation. Here we consider maps
Rijklmnp : Uijklmn × Uijklmp × Uijknp × Uijkmnp × Uijlmnp × Uilmnp × Ujklmnp
−→ Ujklmnp × Uiklmnp × Uijlmnp × Uijkmnp × Uijklnp × Uijklmp × Uijklmn,
where i < j < k < l < m < n < p. The 7-simplex equation reads
R1̂,1R2̂,7P13P14P15P16P17P18P19P20P21P22P23P24P25P26P27P6P5P4P3
P2P7P6P5P4P8P7P6P9P8P10R3̂,11P17P18P19P20P21P22P23P24P25P26
P16P17P18P19P20P21P22P23P24P25P10P9P8P7P6P5P4P3P11P10P9P8
P7P6P12P11P10P9P13P12R4̂,13P19P20P21P22P23P24P25P18P19P20P21
P22P23P17P18P19P20P21P22P12P11P10P9P8P7P6P5P4P13P12P11P10
P9P8P14P13P12R5̂,13P19P20P21P22P23P24P18P19P20P21P17P18P19P16
P17P18P12P11P10P9P8P7P6P5P13P12P11P10R6̂,11P17P18P19P20P21P22
P23P16P17P18P19P15P16P14P13P10P9P8P7P6R7̂,7P13P14P15P16P17P18
P19P20P21P22P12P13P14P15P16P17P11P12P13P10R8̂,1P7P8P9P10P11P12
P13P14P15P16P17P18P19P20P21P6P7P8P9P10P11P12P13P14P15P5P6P7
P8P9P10P4P5P6P3
= P25P22P23P24P21P18P19P20P21P22P23P17P18P19P16P13P14P15P16P17
P18P19P20P21P22P12P13P14P15P16P17P11P12P13P10P7P8P9P10P11
P12P13P14P15P16P17P18P19P20P21P6P7P8P9P10P11P12P13P14P15P5
P6P7P8P9P10P4P5P6P3R8̂,22R7̂,16P22P23P24P25P26P21P22P23P24P20
P21P22P19P20P18P15P14P13P12P11P10P9P8P7P6P5P4P3P2P1R6̂,12P18
P19P20P21P22P23P24P25P17P18P19P20P21P22P16P17P18P19P15P16P11
P10P9P8P7P6P5P4P3P2P12P11P10P9P8P7P6P5P4P3R5̂,10P16P17P18
P19P20P21P22P23P24P15P16P17P18P19P20P14P15P16P9P8P7P6P5P4
P10P9P8P7P6P5P11P10P9P8P7P6R4̂,10P16P17P18P19P20P21P22P23
P15P16P17P18P9P8P7P10P9P8P11P10P9P12P11P10R3̂,12P18P19P20P21
P22P11P12P13P14P15R2̂,16R1̂,22.
Also see Fig. 13. In terms of R̂ = RP35P26P17, it collapses to
R̂1̂,1,2,3,4,5,6,7R̂2̂,1,8,9,10,11,12,13R̂3̂,2,8,14,15,16,17,18R̂4̂,3,9,14,19,20,21,22
R̂5̂,4,10,15,19,23,24,25R̂6̂,5,11,16,20,23,26,27R̂7̂,6,12,17,21,24,26,28R̂8̂,7,13,18,22,25,27,28
= R̂8̂,7,13,18,22,25,27,28R̂7̂,6,12,17,21,24,26,28R̂6̂,5,11,16,20,23,26,27R̂5̂,4,10,15,19,23,24,25
R̂4̂,3,9,14,19,20,21,22R̂3̂,2,8,14,15,16,17,18R̂2̂,1,8,9,10,11,12,13R̂1̂,1,2,3,4,5,6,7.
Remark 3.4. The form in which the above zonohedra appear, i.e., decomposed in two comple-
mentary parts, reveals an interesting feature. If we identify antipodal edges (carrying the same
label) of the boundaries, in each of the two parts, we obtain the same projective polyhedron.13
From the cube, associated with the Yang–Baxter equation, we obtain in this way two copies
of the hemicube. From the permutahedron, associated with the Zamolodchikov equation, we
13Corresponding resolutions of small cubes have to be chosen.
22 A. Dimakis and F. Müller-Hoissen
786867
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34 =
Figure 13. Two complementary sides of a polyhedral reduction of B(8, 5). The 7-simplex equation
corresponds to sequences of maximal chains on them.
obtain two copies of a “hemi-permutahedron”. Here the identification of antipodal edges means
identification of a permutation with the corresponding reversed permutation, e.g., 1234 ∼= 4321,
1324 ∼= 4231.
3.3 Lax systems for simplex equations
We promote the maps RJ : U−→
P (J)
→ U←−
P (J)
, J ∈
([N+2]
N
)
, to “localized” maps
LJ : UJ −→ Map(U−→
P (J)
,U←−
P (J)
),
uJ 7−→ LJ(uJ) : U−→
P (J)
→ U←−
P (J)
.
In B(N + 2, N − 1), counterparts of the two maximal chains, of which B(N + 1, N − 1) consists,
appear as chains for all k̂ = [N + 2] \ {k} = {k1, . . . , kN+1} ∈
([N+2]
N+1
)
:
Ck̂,lex : [αk̂]
k̂kN+1−→ [ρk̂,1]
k̂kN−→ · · · k̂k2−→ [ρk̂,N ]
k̂k1−→ [ωk̂],
Ck̂,rev : [αk̂]
k̂k1−→ [σk̂,1]
k̂k2−→ · · · k̂kN−→ [σk̂,N ]
k̂kN+1−→ [ωk̂],
where αk̂, ωk̂, ρk̂,i, σk̂,i are admissible linear orders of
(
k̂
N−1
)
, the ki are assumed to be in natural
order, and k̂ki = k̂ \ {ki} = [N + 2] \ {k, ki}. For each k ∈ [N + 2], we then impose the localized
N -simplex equation,
LC̃k̂,lex
(u−→
P (k̂)
) = LC̃k̂,rev
(v←−
P (k̂)
),
with u−→
P (k̂)
= (u
k̂kN+1
, . . . , u
k̂k1
) and v←−
P (k̂)
= (v
k̂k1
, . . . , v
k̂kN+1
). We assume that, for each
k ∈ [N + 2], this equation uniquely determines a map Rk̂ : u−→
P (k̂)
7→ v←−
P (k̂)
.
In terms of
L̂
k̂ki
:= L
k̂ki
P
k̂ki
,
Simplex and Polygon Equations 23
the above equation has the form
L̂
k̂k1,XA1
L̂
k̂k2,XA2
· · · L̂
k̂kN+1,XAN+1
=
(
L̂
k̂kN+1,XAN+1
· · · L̂
k̂k2,XA2
L̂
k̂k1,XA1
)
◦ Rk̂. (3.10)
Here XAi = (xai,1 , . . . ,xai,N+1), where 1 ≤ xai,j ≤ c(N + 2, N − 1), are increasing sequences
of positive integers and Ai = (ai,1, . . . ,ai,N+1), where 1 ≤ ai,j ≤ c(N + 1, N − 1), are the
multi-indices introduced in Remark 3.3.
With ρ = (J1, . . . , Jc(N+2,N)) ∈ A(N + 2, N), we associate the composition
L̂ρ = L̂Jc(N+2,N),Ac(N+2,N)
· · · L̂J1,A1 : Uη → Uη
of the corresponding maps. Here η ∈ A(N + 2, N − 1) is the reverse lexicographical order of([N+2]
N−1
)
, and the multi-indices Ai specify the positions of the elements of P (Ji) in η. The next
observation will be important in the following.
Lemma 3.5. If J, J ′ ∈
([N+2]
N
)
satisfy E(J)∩E(J ′) = ∅, then L̂J,AL̂J ′,A′ = L̂J ′,A′L̂J,A (acting
on some Uµ, µ ∈ A(N + 2, N − 1)).
Proof. It is easily verified that E(J) ∩ E(J ′) = ∅ ⇐⇒ P (J) ∩ P (J ′) = ∅. Hence, if E(J) ∩
E(J ′) = ∅, then L̂J,A and L̂J ′,A′ must act on distinct positions in Uµ, hence they commute. �
Now we sketch a proof of the claim that the (N + 1)-simplex equation
RC̃lex
= RC̃rev
,
where C̃lex and C̃rev constitute a resolution of B(N + 2, N), arises as a consistency condition of
the above Lax system. We start with L̂α, where α ∈ A(N + 2, N) is the lexicographical order
of
([N+2]
N
)
, and proceed according to the resolution C̃lex. The above Lemma guarantees that
there is a permutation of L̂’s, corresponding to the resolution of [α] that leads to Uρ0 = Pρ0,αUα,
which arranges that L̂−→
P (N̂+2)
acts on Uρ0 at consecutive positions. This yields L̂ρ0 ◦Pρ0,α. Next
we apply the respective Lax equation (3.10), which results in L̂ρ′0 ◦ (R
N̂+2,a0
Pρ0,α). Proceeding
in this way, we finally arrive at L̂ω ◦ RC̃lex
. Starting again with L̂α, but now following the
resolution C̃rev, we finally obtain L̂ω ◦RC̃rev
. Since we assumed that the Lax equations uniquely
determine the respective maps, we can conclude that the (N + 1)-simplex equation holds.
Example 3.6. Let N = 3. Then α = (4̂5, 3̂5, 3̂4, 2̂5, 2̂4, 2̂3, 1̂5, 1̂4, 1̂3, 1̂2) ∈ A(5, 3) and η =
(45, 35, 34, 25, 24, 23, 15, 14, 13, 12) ∈ A(5, 2), from which we can read off the position indices to
obtain
L̂α = L̂1̂2,123L̂1̂3,145L̂1̂4,246L̂1̂5,356L̂2̂3,178L̂2̂4,279L̂2̂5,389L̂3̂4,470L̂3̂5,580L̂4̂5,690.
The Lax system takes the form
L̂
k̂k1,x1,x2,x3
L̂
k̂k2,x1,x4,x5
L̂
k̂k3,x2,x4,x6
L̂
k̂k4,x3,x5,x6
=
(
L̂
k̂k4,x3,x5,x6
L̂
k̂k3,x2,x4,x6
L̂
k̂k2,x1,x4,x5
L̂
k̂k1,x1,x2,x3
)
◦ Rk̂,
where 1 ≤ x1 < x2 < · · · < x6 ≤ 10. Now we have
L̂α = L̂1̂2,123L̂1̂3,145L̂1̂4,246L̂1̂5,356L̂2̂3,178L̂2̂4,279L̂2̂5,389L̂3̂4,470L̂3̂5,580L̂4̂5,690
P4P5P6P3= L̂1̂2,123L̂1̂3,145L̂1̂4,246L̂2̂3,178L̂2̂4,279L̂3̂4,470L̂1̂5,356L̂2̂5,389L̂3̂5,580L̂4̂5,690
24 A. Dimakis and F. Müller-Hoissen
R5̂,1
= L̂1̂2,123L̂1̂3,145L̂1̂4,246L̂2̂3,178L̂2̂4,279L̂3̂4,470L̂4̂5,690L̂3̂5,580L̂2̂5,389L̂1̂5,356
P7= L̂1̂2,123L̂1̂3,145L̂2̂3,178L̂1̂4,246L̂2̂4,279L̂3̂4,470L̂4̂5,690L̂3̂5,580L̂2̂5,389L̂1̂5,356
R4̂,4
= L̂1̂2,123L̂1̂3,145L̂2̂3,178L̂4̂5,690L̂3̂4,470L̂2̂4,279L̂1̂4,246L̂3̂5,580L̂2̂5,389L̂1̂5,356
P8P7P4P3= L̂1̂2,123L̂4̂5,690L̂1̂3,145L̂2̂3,178L̂3̂4,470L̂3̂5,580L̂2̂4,279L̂1̂4,246L̂2̂5,389L̂1̂5,356
R3̂,5
= L̂1̂2,123L̂4̂5,690L̂3̂5,580L̂3̂4,470L̂2̂3,178L̂1̂3,145L̂2̂4,279L̂1̂4,246L̂2̂5,389L̂1̂5,356
P7P8P9P3P2P4= L̂4̂5,690L̂3̂5,580L̂3̂4,470L̂1̂2,123L̂2̂3,178L̂2̂4,279L̂2̂5,389L̂1̂3,145L̂1̂4,246L̂1̂5,356
R2̂,4
= L̂4̂5,690L̂3̂5,580L̂3̂4,470L̂2̂5,389L̂2̂4,279L̂2̂3,178L̂1̂2,123L̂1̂3,145L̂1̂4,246L̂1̂5,356
R1̂,1
= L̂4̂5,690L̂3̂5,580L̂3̂4,470L̂2̂5,389L̂2̂4,279L̂2̂3,178L̂1̂5,356L̂1̂4,246L̂1̂3,145L̂1̂2,123,
where an index 0 stands for 10 (ten). Here we indicated over the equality signs the maps that
act on the arguments of the L̂’s in the respective transformation step. Returning to the proper
notation, the result is L̂α = L̂ω ◦ RC̃lex
, which determines the left hand side of the 4-simplex
equation (3.6). We marked in brown those L̂’s that have to be brought together in order to
allow for an application of a Lax equation. We could have omitted all the position indices in the
above computation, since they are automatically compatible according to Lemma 3.5. But we
kept them for comparison with corresponding computations in the literature (see, e.g., [44, 73,
74, 75, 77, 82]), where only these position indices appear, but not the “combinatorial indices”
that nicely guided us through the above computation.
3.4 Reductions of simplex equations
The relation between the Bruhat order B(N + 1, N − 1) and the N -simplex equation, together
with the projection of Bruhat orders, defined in Remark 2.5, induces a relation between neigh-
boring simplex equations:
B(N + 2, N) ←→ (N + 1)-simplex equation
↓ ↓
B(N + 1, N − 1) ←→ N -simplex equation
Since the structure of the N -simplex equation can be read off in full detail from B(N+1, N−2),
we shall consider the projection B(N + 2, N − 1)→ B(N + 1, N − 2). Let us choose k = N + 2
in Remark 2.5. Then all vertices of B(N + 2, N − 1) connected by edges labeled by ̂j,N+2 (in
complementary notation), with some j < N + 2, are identified under the projection.
For the example of B(4, 1) and k = 4, the projection is shown in Fig. 3. The 3-simplex
equation
R̂(3)
1̂,123
R̂(3)
2̂,145
R̂(3)
3̂,246
R̂(3)
4̂,356
= R̂(3)
4̂,356
R̂(3)
3̂,246
R̂(3)
2̂,145
R̂(3)
1̂,123
acts on U12×U13×U14×U23×U24×U34. The projection formally reduces this to the 2-simplex
equation
R̂(2)
1̂,12
R̂(2)
2̂,13
R̂(2)
3̂,23
= R̂(2)
3̂,23
R̂(2)
2̂,13
R̂(2)
1̂,12
,
acting on U12 × U13 × U23. Of course, in the two equations we are dealing with different types
of maps (indicated by a superscript (2), respectively (3)). The following relation holds.
Simplex and Polygon Equations 25
Proposition 3.7. Let R̂(N) satisfy the N -simplex equation. Let U
ĵ,N+2
, j = 1, 2, . . . , N + 1, be
sets, fj : U
ĵ,N+2
→ U
ĵ,N+2
and R̂(N+1)
N̂+2
: U
1̂,N+2
×U
2̂,N+2
× · · · ×U ̂N+1,N+2
→ U
1̂,N+2
×U
2̂,N+2
×
· · · × U ̂N+1,N+2
any maps such that
(f1 × f2 × · · · × fN+1) R̂(N+1)
N̂+2
= R̂(N+1)
N̂+2
(f1 × f2 × · · · × fN+1)
holds. Setting
R̂(N+1)
ĵ
:= R̂(N)
ĵ
× fj , j = 1, 2, . . . , N + 1,
then yields a solution R̂(N+1)
ĵ
, j = 1, . . . , N + 2, of the (N + 1)-simplex equation.
Proof. The statement can be easily verified by a direct computation. �
In particular, if the maps fj , j = 1, 2, . . . , N + 1, are identity functions, then the condition
in the proposition is trivially satisfied and the new map defined in terms of the N -simplex map
solves the (N + 1)-simplex equation.
4 Polygon equations
In this section we address realizations of Tamari orders T (N,n) in terms of sets and maps
between Cartesian14 products of these sets. After some preparations in Section 4.1, polygon
equations will be introduced in Section 4.2, which contains explicit expressions up to the 11-
gon equation, and the associated polyhedra. Section 4.3 discusses the integrability of polygon
equations. Reductions of polygon equations associated with reductions of Tamari orders are
the subject of Section 4.4. In the following, we write ρ̄ instead of ρ(b), for an admissible linear
order ρ.
4.1 Resolutions of T (N,N − 2) and polyhedra
The Tamari order T (N,N − 2) consists of the two maximal chains
Co : [ᾱ]
N̂−→ [ρ̄1]
N̂−2−→ [ρ̄3] −→ · · · −→ [ρ̄N+m−3]
2̂−m−→ [ω̄],
Ce : [ᾱ]
1̂+m−→ [σ̄1+m]
3̂+m−→ [σ̄3+m] −→ · · · −→ [σ̄N−5]
N̂−3−→ [σ̄N−3]
N̂−1−→ [ω̄],
where m = N mod 2. There are resolutions C̃o and C̃e in A(b)(N,N − 2), both starting with ᾱ
and both ending with ω̄,
ᾱ ω̄
C̃o
C̃e
Using the correspondence between elements of A(b)(N,N−2) and maximal chains of T (N,N−3)
(see Remark 2.16), each of the two resolutions corresponds to a sequence of maximal chains
of T (N,N−3). For ρ̄ ∈ A(b)(N,N−2), let Cρ̄ be the corresponding maximal chain of T (N,N−3).
The resolution C̃o, respectively C̃e, is then a rule for deforming Cᾱ stepwise into Cω̄.
14If the sets are supplied with a linear structure, we may as well consider tensor products or direct sums.
26 A. Dimakis and F. Müller-Hoissen
The resolution of T (N,N − 2), given by C̃o and C̃e, can be regarded as a rule to construct
a polyhedron. If N is odd, i.e., N = 2n + 1, then ᾱ and ω̄ have both n(n + 1)/2 elements.
The construction rules are then exactly the same as in the case treated in Section 3.1. Each
appearance of an inversion corresponds to a 2n-gon.
If N is even, i.e., N = 2n, then ᾱ has n(n + 1)/2 elements and ω̄ has n(n − 1)/2 elements.
Starting from the top vertex, the chain corresponding to ᾱ (ω̄) is drawn counterclockwise (clock-
wise). The two chains ᾱ and ω̄ then join to form an n2-gon. Again, the two sides of the N -gon
equation correspond to sequences of maximal chains that deform ᾱ into ω̄. But in this case we
do not obtain a zonohedron, since an inversion is represented by a (2n − 1)-gon, hence an odd
polygon.
4.2 Polygon equations and associated polyhedra
Let N ∈ N, N > 1, and 0 ≤ n ≤ N − 1. With ρ̄ ∈ A(b)(N,n) we associate the corresponding
Cartesian product Uρ̄ of sets UJ , J ∈ ρ̄. For each K ∈
(
[N ]
n+1
)
, let there be a map
TK : U−→
Po(K)
→ U←−
Pe(K)
,
where
−→
Po(K) and
←−
Pe(K) have been defined in Section 2.2, and U−→
Po(K)
, U←−
Pe(K)
are the corre-
sponding Cartesian products, i.e.,
U−→
Po(K)
:= UK\{kn+1} × UK\{kn−1} × · · · × UK\{k1+(nmod 2)},
U←−
Pe(K)
:= UK\{k2−(nmod 2)} × · · · × UK\{kn−2} × UK\{kn}.
Remark 4.1. In case of the dual B(r)(N,n) of the Tamari order T (N,n), we are dealing instead
with maps SK : U−→
Pe(K)
−→ U←−
Po(K)
.
Let [ρ̄]
K−→ [ρ̄′], K ∈
(
[N ]
n+1
)
, be an inversion in T (N,n). Hence
−→
Po(K) ⊂ ρ̄ and
←−
Pe(K) ⊂ ρ̄′
(where ⊂ means subsequence). If
−→
Po(K) appears in ρ̄ at consecutive positions, starting at
position a, we extend TK to a map TK,a : Uρ̄ → Uρ̄′ , which acts non-trivially only on the sets
labeled by the elements of Po(K).
For a maximal chain C : [ᾱ]
K1−→ [ρ̄1]
K2−→ [ρ̄2] −→ · · · Kr−→ [ω̄] of T (N,n), let C̃ be a resolution
of C in A(b)(N,n). We write TC̃ : Uᾱ → Uω̄ for the corresponding composition of maps TKi,ai
and Pa.15 Turning to T (N,N − 2) and choosing α as the lexicographically ordered set
( [N ]
N−2
)
and ω as α in reverse order, the N -gon equation is defined by
TC̃o = TC̃e ,
which is independent of the choice of resolutions.
For odd N , i.e., N = 2n + 1, ᾱ is the lexicographically ordered sequence of elements
̂(2j)(2k + 1), with j = 1, . . . , n and k = j, . . . , n. ω̄ is the reverse lexicographically ordered se-
quence of elements ̂(2j − 1)(2k), where j = 1, . . . , n and k = j, . . . , n. Here Tk̂ : U−→
Po(k̂)
→ U←−
Pe(k̂)
acts between Cartesian products having n factors. In terms of16
T̂k̂ = Tk̂Pk̂ : U←−
Po(k̂)
→ U←−
Pe(k̂)
,
15TC̃ has the form Pω̄,ρ̄′rTKr,arPρ̄r−1,ρ̄
′
r−1
· · · Pρ̄2,ρ̄′2TK2,a2Pρ̄1,ρ̄′1TK1,a1Pρ̄0,ᾱ. Here, for i = 1, . . . , r, ρ̄i−1 ∈ [ρ̄i−1]
(where ρ̄0 ∈ [ᾱ]) such that
−→
Po(Ki) appears in it at consecutive positions, starting at ai. ρ̄
′
i ∈ [ρ̄i] is the result of
applying TKi,ai to ρ̄i (so that ρ̄′i contains
←−
Pe(Ki) at consecutive positions).
16The permutation map Pk̂ achieves a reversion.
Simplex and Polygon Equations 27
the two sides of the (2n+ 1)-gon equation become maps Urev(ᾱ) → Uω̄, where rev(ᾱ) is ᾱ reverse
lexicographically ordered. ᾱ and ω̄ have both c(n + 1, 2) = 1
2n(n + 1) elements. The “hatted
polygon equation” can be obtained either by substituting Tk̂ = T̂k̂Pk̂ in the original polygon
equation, or by starting with Urev(ᾱ) and stepwise mapping it to Uω̄, following Co and Ce. It has
the form
T̂1̂,B1
T̂3̂,B3
· · · T̂
2̂n+1,B2n+1
= T̂2̂n,B2n
· · · T̂4̂,B4
T̂2̂,B2
,
where Bk = (bk,1, . . . , bk,n), with 1 ≤ bk,i ≤ c(n+ 1, 2), is the multi-index (increasing sequence
of positive integers) specifying the positions, in the respectice active linear order, which take
part in the action of the map T̂k̂. Examples will be presented below.
For even N , i.e., N = 2n, ᾱ is the lexicographically ordered sequence ̂(2j − 1)(2k), where j =
1, . . . , n and k = j, . . . , n, and ω̄ is the reverse lexicographically ordered sequence ̂(2j)(2k + 1),
j = 1, . . . , n−1 and k = j, . . . , n−1. Thus, ᾱ has c(n+1, 2) and ω̄ has c(n, 2) elements. Now Tk̂
maps U−→
Po(k̂)
, which has n factors, to U←−
Pe(k̂)
, which has n−1 factors. Also in this case the polygon
equation can be expressed in compact form without the need of permutation maps. But in order
to achieve this, we have to modify the range to U
0̂k
× U←−
Pe(k̂)
, where the sets U
0̂k
play the role
of placeholders, they are irrelevant for the process of evaluation of the polygon equation. We
define
T̂k̂ := (u
0̂k
, Tk̂Pk̂) : U←−
Po(k̂)
→ U
0̂k
× U←−
Pe(k̂)
,
choosing fixed elements u
0̂k
∈ U
0̂k
. The 2n-gon equation then takes the form
T̂2̂,B2
T̂4̂,B4
· · · T̂2̂n,B2n
= T̂
2̂n−1,B2n−1
· · · T̂3̂,B3
T̂1̂,B1
, (4.1)
where Bk = (bk,1, . . . , bk,n), 1 ≤ bk,i ≤ c(n + 1, 2). This requires setting U
0̂ (2l)
= U ̂0 (2l−1)
for l = 1, . . . , n. Then both sides of (4.1) are maps Urev(ᾱ) → U0̂1 × · · · × U ̂0 (2l−1)
× Uω̄ =
U0̂2 × · · · × U0̂ (2l)
× Uω̄.
Digon equation. The two maximal chains of T (2, 0) lead to the digon equation T1 = T2 for
the two maps Ti : U∅ → U∅.
Trigon equation. The two maximal chains of T (3, 1) are 1
12→ 2
23→ 3 and 1
13→ 3. The maps
Tij : Ui → Uj , i < j, have to satisfy the trigon equation T23T12 = T13.
Tetragon equation. The two maximal chains of T (4, 2) are
12
23
34
123−→ 13
34
134−→ 14
12
23
34
234−→ 12
14
124−→ 14
The tetragon equation is thus
T134T123,1 = T124T234,2,
for maps Tijk : Uij×Ujk → Uik, i < j < k. Using complementary notation, the tetragon equation
reads
T2̂T4̂,1 = T3̂T1̂,2.
28 A. Dimakis and F. Müller-Hoissen
23
24
34
1214
2334
13
14 12
=
Figure 14. Graphical representation of the tetragon equation (using complementary notation for the
edge labels, with hats omitted).
45
25
23
15
35 13
34
12
14
45
25
23
12
24
12
45
14
34
=
Figure 15. Two complementary sides of the cube formed by T (5, 2). We use complementary notation for
the labels. The equality presents the pentagon equation in shorthand form. See Fig. 2 for the expanded
form (using original labeling).
Also see Fig. 14. The hatted version of the tetragon equation is
T̂2̂,13T̂4̂,23 = T̂3̂,23T̂1̂,12,
which can be read off from
1̂2
1̂4
3̂4
4̂−→
23
1̂2
0̂4
2̂4
2̂−→
13
0̂2
0̂4
2̂3
1̂2
1̂4
3̂4
1̂−→
12
0̂1
1̂3
3̂4
3̂−→
23
0̂1
0̂3
2̂3
As here, also in the following we will sometimes superfluously display read-off position indices
under the arrows.
Pentagon equation. The two maximal chains of T (5, 3) can be resolved to
123
134
145
1234−→
234
124
145
1245−→
234
245
125
2345−→
345
235
125
123
134
145
1345−→
123
345
135
∼−→
345
123
135
1235−→
345
235
125
They describe deformations of maximal chains of T (5, 2), see Fig. 2. Here we consider maps
Tijkl : Uijk × Uikl → Ujkl × Uijl, i < j < k < l. Using complementary notation, the pentagon
equation is thus
T1̂,1T3̂,2T5̂,1 = T4̂,2P1T2̂,2, (4.2)
also see Fig. 15. In terms of T̂ := T P, it takes the form
T̂1̂,12T̂3̂,13T̂5̂,23 = T̂4̂,23T̂2̂,12. (4.3)
Hexagon equation. We will treat this case in some more detail. The two maximal chains of
T (6, 4) are
Co : [ᾱ]
12345−→ [ρ̄1]
12356−→ [ρ̄3]
13456−→ [ω̄], Ce : [ᾱ]
23456−→ [σ̄1]
12456−→ [σ̄3]
12346−→ [ω̄],
Simplex and Polygon Equations 29
Figure 16. The left-hand side of the hexagon equation (4.4) corresponds to a sequence of maximal chains
(first row) on one side of the associahedron in three dimensions, formed by the Tamari lattice T (6, 3).
The right-hand side corresponds to a sequence of maximal chains (second row) on the complementary
side.
with ᾱ = (1234, 1245, 1256, 2345, 2356, 3456). The equivalence class [ᾱ] contains another linear
order, which is (1234, 1245, 2345, 1256, 2356, 3456). We obtain
C̃o :
1234
1245
1256
2345
2356
3456
∼−→
1234
1245
2345
1256
2356
3456
12345−→
1345
1235
1256
2356
3456
12356−→
1345
1356
1236
3456
∼−→
1345
1356
3456
1236
13456−→
1456
1346
1236
C̃e :
1234
1245
1256
2345
2356
3456
23456−→
1234
1245
1256
2456
2346
12456−→
1234
1456
1246
2346
∼−→
1456
1234
1246
2346
12346−→
1456
1346
1236
In this case, we consider maps Tijklm : Uijkl×Uijlm×Ujklm → Uiklm×Uijkm, i < j < k < l < m.
Using complementary notation, the chains C̃o and C̃e read17
5̂6
3̂6
3̂4
1̂6
1̂4
1̂2
∼−→
34
5̂6
3̂6
1̂6
3̂4
1̂4
1̂2
6̂−−→
123
2̂6
4̂6
3̂4
1̂4
1̂2
4̂−−→
234
2̂6
2̂4
4̂5
1̂2
∼−→
34
2̂6
2̂4
1̂2
4̂5
2̂−−→
123
2̂3
2̂5
4̂5
5̂6
3̂6
3̂4
1̂6
1̂4
1̂2
1̂−−→
456
5̂6
3̂6
3̂4
1̂3
1̂5
3̂−−→
234
5̂6
2̂3
3̂5
1̂5
∼−→
12
2̂3
5̂6
3̂5
1̂5
5̂−−→
234
2̂3
2̂5
4̂5
This corresponds to the two sequences of graphs in Fig. 16. We read off the hexagon equation
T2̂,1P3T4̂,2T6̂,1P3 = T5̂,2P1T3̂,2T1̂,4. (4.4)
Fig. 17 is a short-hand form of Fig. 16. In a categorical setting, a similar diagram appeared
in [50, p. 218], and in [94, p. 189] as a 4-cycle condition. According to the prescription given
for even polygon equations in the beginning of this subsection, we obtain the following hatted
version of the hexagon equation,
T̂2̂,145T̂4̂,246T̂6̂,356 = T̂5̂,356T̂3̂,245T̂1̂,123, (4.5)
17Again, the boldface digits are positions, counted from top to bottom in a column, on which the corresponding
action takes place. Here they are always consecutive and can thus be abbreviated to the first, as done in (4.4).
30 A. Dimakis and F. Müller-Hoissen
23
26
56
25
2446
36
45
12
4534
16
34
12
14
16
2356
25
5623
36
45
35
34
1513
16 12
14
=
Figure 17. Two complementary sides of the associahedron. The equality expresses the hexagon equation.
67
47
45
27
25
23
27
45
17
37
57
15
35
56
15
37
23
56
13
34
36
12
14
16
67
47
45
27
25
23
12
24
26
12
45
12
47 14
34
46
12
67
14
67
34
67
16
36
56
16
34
=
Figure 18. Two complementary sides of the Edelman–Reiner polyhedron, formed by T (7, 4). The
equality represents the heptagon equation.
which can be read off from
1̂2
1̂4
1̂6
3̂4
3̂6
5̂6
6̂−−→
356
1̂2
1̂4
0̂6
3̂4
2̂6
4̂6
4̂−−→
246
1̂2
0̂4
0̂6
2̂4
2̂6
4̂5
2̂−−→
145
0̂2
0̂4
0̂6
2̂3
2̂5
4̂5
1̂2
1̂4
1̂6
3̂4
3̂6
5̂6
1̂−−→
123
0̂1
1̂3
1̂5
3̂4
3̂6
5̂6
3̂−−→
245
0̂1
0̂3
1̂5
2̂3
3̂5
5̂6
5̂−−→
356
0̂1
0̂3
0̂5
2̂3
2̂5
4̂5
Some versions of (4.5) appeared in [54, 61, 62, 63, 64, 65, 66] as a “Pachner relation” for a map
realizing Pachner moves of triangulations of a four-dimensional manifold.
Heptagon equation. Here we consider maps Tijklmp : Uijklm × Uijkmp × Uiklmp → Ujklmp ×
Uijlmp × Uijklp, i < j < k < l < m < p. The two maximal chains of T (7, 5) lead to the heptagon
equation
T1̂,1T3̂,3P5P2T5̂,3T7̂,1P3 = P3T6̂,4P3P2P1T4̂,3P2P3T2̂,4, (4.6)
using complementary notation. Fig. 18 shows the two sides of the Edelman–Reiner polyhe-
dron [30], formed by T (7, 4), on which the respective sides of this equation correspond to se-
quences of maximal chains. In terms of T̂ := T P13, the heptagon equation takes the form
T̂1̂,123T̂3̂,145T̂5̂,246T̂7̂,356 = T̂6̂,356T̂4̂,245T̂2̂,123. (4.7)
An equation with this structure appeared in [100].
Octagon equation. In case of T (8, 6), we consider maps Tijklmpq : Uijklmp×Uijklpq×Uijlmpq×
Ujklmpq → Uiklmpq × Uijkmpq × Uijklmp, i < j < k < l < m < p < q, subject to the octagon
equation
T2̂,1P4P5P6T4̂,3P6P5P2T6̂,3P6T8̂,1P4P5P6P3 = P3T7̂,4P3P2P1T5̂,3P6P2P3T3̂,4T1̂,7.
Simplex and Polygon Equations 31
23
28
78
25
2648
58
27
24
4826
68
38
56
45
12
45
46
56
185638
47
12
47
3467
36
18
36
67
12
67
14
6734
16
34
18
34
12
1416
18
2378
25782358
27
45
78255823
56
45
27
78
4556
23
38
47
5735
36
67
37
15
37
34
17
15
13
18 12
1416
=
Figure 19. Two complementary sides of the polyhedron formed by T (8, 5). Equality represents the
octagon equation.
See Fig. 19. The hatted version of the octagon equation is
T̂2̂,1,5,6,7T̂4̂,2,5,8,9T̂6̂,3,6,8,10T̂8̂,4,7,9,10 = T̂7̂,4,7,9,10T̂5̂,3,6,8,9T̂3̂,2,5,6,7T̂1̂,1,2,3,4.
The position indices can be read off from
1̂2
1̂4
1̂6
1̂8
3̂4
3̂6
3̂8
5̂6
5̂8
7̂8
8̂−−−−−→
4,7,9,10
1̂2
1̂4
1̂6
0̂8
3̂4
3̂6
2̂8
5̂6
4̂8
6̂8
6̂−−−−−→
3,6,8,10
1̂2
1̂4
0̂6
0̂8
3̂4
2̂6
2̂8
4̂6
4̂8
6̂7
4̂−−−−→
2,5,8,9
1̂2
0̂4
0̂6
0̂8
2̂4
2̂6
2̂8
4̂5
4̂7
6̂7
2̂−−−−→
1,5,6,7
0̂2
0̂4
0̂6
0̂8
2̂3
2̂5
2̂7
4̂5
4̂7
6̂7
1̂2
1̂4
1̂6
1̂8
3̂4
3̂6
3̂8
5̂6
5̂8
7̂8
1̂−→
0̂2
1̂3
1̂5
1̂7
3̂4
3̂6
3̂8
5̂6
5̂8
7̂8
3̂−→
0̂1
0̂3
1̂5
1̂7
2̂3
3̂5
3̂7
5̂6
5̂8
7̂8
5̂−→
0̂1
0̂3
0̂5
1̂7
2̂3
2̂5
3̂7
4̂5
5̂7
7̂8
7̂−→
0̂1
0̂3
0̂5
0̂7
2̂3
2̂5
2̂7
4̂5
4̂7
6̂7
Enneagon equation. For T (9, 7) we find
T1̂,1T3̂,4P7P8P9P3P2P4T5̂,5P8P7P4P3T7̂,4P7T9̂,1P4P5P6P3
= P7P4P5P6P3T8̂,7P6P5P4P3P2P1T6̂,5P8P4P3P2P5P4P3T4̂,5P4P5P6T2̂,7,
which can be visualized on T (9, 6), see Fig. 20. In terms of T̂ := T P14P23, the enneagon (or
nonagon) equation takes the compact form
T̂1̂,1,2,3,4T̂3̂,1,5,6,7T̂5̂,2,5,8,9T̂7̂,3,6,8,10T̂9̂,4,7,9,10 = T̂8̂,4,7,9,10T̂6̂,3,6,8,9T̂4̂,2,5,6,7T̂2̂,1,2,3,4.
Decagon equation. For T (10, 8) we obtain the equation
T2̂,1P5P6P7P8P9P10T4̂,4P8P9P10P7P8P9P3P2P4T6̂,5P9P10P8P7
P4P3T8̂,4P8P9P10P7T1̂0,1P5P6P7P8P9P10P4P5P6P3
= P7P4P5P6P3T9̂,7P6P5P4P3P2P1T7̂,5P9P10P8P4P3P2P5P4P3T5̂,5
P9P10P4P5P6T3̂,7T1̂,11,
32 A. Dimakis and F. Müller-Hoissen
89
69
67
49
47
45
29
27
25
23
49
67
29
45
29
47
29
67
19
39
59
79
27
45
17
37
57
78
17
59
37
59
45
78 25
78
15
35
56
58
15
37
17
39
15
39
23
78
23
58
23
56
13
34
36
38
12
14
16
18
89
69
67
49
47
45
29
27
25
23
12
24
26
28
12
45
12
47
12
49
14
34
46
48
12
67 14
67 34
67
12
69 14
69
34
69
26
48
16
36
56
68
12
89 14
89
34
89
16
89
36
89
56
89
18
38
58
78
16
34
18
56
18
36
18
34
38
56
=
Figure 20. Two complementary sides of the polyhedral part of T (9, 6) (which is non-polyhedral due to
the existence of small cubes). Equality represents the enneagon equation.
23
2A
9A
25
284A
7A
27
266A
5A
78
29
24
4A288A
3A
5A
45
12
45
48
266A
48
78
1A58
47
47
4668
58
56
49
49
34
6756
89
38
3A
6714
6936
18
38
69
16
36
8934
12
141618
1A
239A
257A
2778
295A
45277A255A
58
47
47
7845
58
56
49
299A
67
5756
23
3A
6749
79
375935
38
69
59
1715
37
36
89
3939
34 19
1715
13
1A 12
141618
=
Figure 21. Two complementary sides of the polyhedral part of T (10, 7) (small cubes are resolved).
Equality represents the decagon equation. Here A stands for 10.
and Fig. 21 shows the corresponding polyhedral representation obtained from T (10, 7). The
hatted version of the decagon equation is
T̂2̂,1,6,7,8,9T̂4̂,2,6,10,11,12T̂6̂,3,7,10,13,14T̂8̂,4,8,11,13,15T̂1̂0,5,9,12,14,15
= T̂9̂,5,9,12,14,15T̂7̂,4,8,11,13,14T̂5̂,3,7,10,11,12T̂3̂,2,6,7,8,9T̂1̂,1,2,3,4,5.
Hendecagon equation. For T (11, 9), the associated equation reads
T1̂,1T3̂,5P9P10P11P12P13P14P4P3P2P5P4P6T5̂,7P11P12P13P10P11P12
P6P5P4P3P7P6T7̂,7P11P12P10P9P6P5P4T9̂,5P9P10P11P8T1̂1,1
P5P6P7P8P9P10P4P5P6P3
= P12P9P10P11P8P5P6P7P8P9P10P4P5P6P3T1̂0,11P10P9P8P7P6P5P4P3P2P1
T8̂,8P12P13P11P7P6P5P4P3P2P8P7P6P5P4P3T6̂,7P11P12P6P5P4P7P6P5
P8P7P6T4̂,8P7P8P9P10T2̂,11,
Simplex and Polygon Equations 33
AB8B
89
6B
69
67
4B
49
47
45
2B
29
27
25
23
6B
4B
69
2B
89
1B
3B
5B
7B
9B
49
29
19
39
59
79
9A
7B
67
47
27
17
37
57
78
7A
59
5B
45
25
15
35
56
58
5A
37
39
19
17
15
3B
9A
7A
78
5A
58
23
56
13
34
36
38
3A
12
14
16
18
1A
AB
8B
89
6B
69
67
4B
49
47
45
2B
29
27
25
23
12
24
26
28
2A
45
47
49
4B
14
34
46
48
4A
67
69
6B
26
16
36
56
68
6A
89
8B
48
4A
28
6A
18
38
58
78
8A
12
14
AB
1A
3A
5A
7A
9A
16
18
36
1A
34
38
58
3A
56
5A
78
=
Figure 22. Two complementary sides of the polyhedral part of T (11, 8) (small cubes are resolved).
Equality represents the hendecagon equation. Here we set A := 10, B := 11.
also see Fig. 22. In terms of T̂ = T P15P24, it takes the form
T̂1̂,1,2,3,4,5T̂3̂,1,6,7,8,9T̂5̂,2,6,10,11,12T̂7̂,3,7,10,13,14T̂9̂,4,8,11,13,15T̂1̂1,5,9,12,14,15
= T̂1̂0,5,9,12,14,15T̂8̂,4,8,11,13,14T̂6̂,3,7,10,11,12T̂4̂,2,6,7,8,9T̂2̂,1,2,3,4,5.
Remark 4.2. Disregarding the indices that specify on which sets the maps act, the left-hand
side of the N -simplex equation has the same structure as the left-hand side of the (2N + 1)-gon
equation (but there is no such relation between the right-hand sides, of course). Accordingly,
the corresponding halfs of the associated polyhedra coincide up to the labeling.
4.3 Lax systems for polygon equations
In this subsection we consider the case where the maps TJ , J ∈
([N+1]
N−1
)
, are “localized” to maps
LJ : UJ −→ Map(U−→
Po(J)
,U←−
Pe(J)
),
uJ 7−→ LJ(uJ) : U−→
Po(J)
→ U←−
Pe(J)
.
In T (N + 1, N − 2), counterparts of the two maximal chains, of which T (N,N − 2) consists,
appear as chains for all k̂ ∈
([N+1]
N
)
, k ∈ [N + 1],
Ck̂,o : [ᾱk̂]
k̂kN−→ [ρ̄k̂,1]
k̂kN−2−→ [ρ̄k̂,2] −→ · · · −→ [ρ̄k̂,N+m−3]
k̂k2−m−→ [ω̄k̂],
Ck̂,e : [ᾱk̂]
k̂k1+m−→ [σ̄k̂,1]
k̂k3+m−→ [σ̄k̂,2] −→ · · · k̂kN−1−→ [ω̄k̂] ,
where we wrote k̂ = (k1, . . . , kN ), k1 < k2 < · · · < kN . Here ᾱk̂, ω̄k̂, ρ̄k̂,i and σ̄k̂,i are reduced
admissible linear orders of
(
k̂
N−2
)
, and m := N mod 2. Let C̃k̂,o and C̃k̂,e be resolutions of the
above chains. We consider the following system of localized N -gon equations,
LC̃k̂,o(u−→Po(k̂)
) = LC̃k̂,e(v←−Pe(k̂)
), k̂ ∈
(
[N + 1]
N
)
, (4.8)
34 A. Dimakis and F. Müller-Hoissen
where
u−→
Po(k̂)
=
(
u
k̂kN
, u
k̂kN−2
, . . . , u
k̂k2−m
)
, v←−
Pe(k̂)
=
(
v
k̂k1+m
, . . . , v
k̂kN−3
, . . . , v
k̂kN−1
)
. (4.9)
We shall assume that each of these equations uniquely determines a map Tk̂ via u−→
Po(k̂)
7→ v←−
Pe(k̂)
.
A hatted version of LJ is defined in the same way as the hatted version of TJ (see the
beginning of Section 4), differently for even and odd polygon equations. The above system (4.8)
then has the form
L̂
k̂k2−m,Y B2−m
· · · L̂
k̂kN−2,Y BN−2
L̂
k̂kN ,Y BN
=
(
L̂
k̂kN−1,Y BN−1
L̂
k̂kN−3,Y BN−3
· · · L̂
k̂km+1,Y Bm+1
)
◦ Tk̂,
where k̂ ∈
([N+1]
N
)
. Here Y Bi = (ybi,1 , . . . ,ybi,n), where n = bN/2c = (N−m)/2, is an increasing
sequence of integers, and Bi = (bi,1, . . . , bi,n) is a multi-index, as defined previously. We have
1 ≤ bi,j ≤ c(n+ 1, 2) and 1 ≤ ybi,j ≤ c(n+m+ 1, 3).
With ρ̄ = (J1, . . . , Jr) ∈ A(b)(N + 1, N − 1) we associate the composition of maps L̂ρ̄ =
L̂Jr,Br · · · L̂J1,B1 . The domain of L̂ρ̄ is Urev(η̄), where η ∈ A(N + 1, N − 1) is the lexicographical
order of
([N+1]
N−2
)
. B1 is the multi-index of the positions of the elements of P (J1) in rev(η̄) (which
has c(n + m + 1, 3) elements). It seems to be a difficult task to find a general formula that
determines the other multi-indices.
In a similar way as in the case of simplex equations, one can show that the (N + 1)-gon
equation TC̃o = TC̃e , where C̃o and C̃e constitute a resolution of T (N + 1, N − 1), arises as
a consistency condition of the above Lax system. Here one starts with L̂ᾱ, ᾱ ∈ A(b)(N+1, N−1),
where α is the lexicographical order of
([N+1]
N−1
)
, and follows the two resolutions.
Example 4.3. For N = 6, we have
rev(η̄) = (1̂23, 1̂25, 1̂27, 1̂45, 1̂47, 1̂67, 3̂45, 3̂47, 3̂67, 5̂67)
and ᾱ = (6̂7, 4̂7, 4̂5, 2̂7, 2̂5, 2̂3) ∈ A(b)(7, 5), so that
1̂23
1̂25
1̂27
1̂45
1̂47
1̂67
3̂45
3̂47
3̂67
5̂67
6̂7−−→
690
1̂23
1̂25
1̂27
1̂45
1̂47
0̂67
3̂45
3̂47
2̂67
4̂67
4̂7−−→
580
1̂23
1̂25
1̂27
1̂45
0̂47
0̂67
3̂45
2̂47
2̂67
4̂57
4̂5−−→
470
1̂23
1̂25
1̂27
0̂45
0̂47
0̂67
2̂45
2̂47
2̂67
4̂56
2̂7−−→
389
1̂23
1̂25
0̂27
0̂45
0̂47
0̂67
2̂45
2̂37
2̂57
4̂56
2̂5−−→
279
1̂23
0̂25
0̂27
0̂45
0̂47
0̂67
2̂35
2̂37
2̂56
4̂56
2̂3−−→
178
0̂23
0̂25
0̂27
0̂45
0̂47
0̂67
2̂34
2̂36
2̂56
4̂56
from which we can read off the position (i.e., boldface) indices of L̂ᾱ. The Lax system reads
L̂ ˆkk2,y1,y4,y5
L̂ ˆkk4,y2,y4,y6
L̂ ˆkk6,y3,y5,y6
= L̂ ˆkk5,y3,y5,y6
L̂ ˆkk3,y2,y4,y5
L̂ ˆkk1,y1,y2,y3
,
where 1 ≤ yb ≤ 10. The consistency condition is now obtained from18
L̂ᾱ = L̂2̂3,178L̂2̂5,279L̂2̂7,389L̂4̂5,470L̂4̂7,580L̂6̂7,690
18Here we depart from our notation and indicate over an equality sign the maps that act on the arguments of
the L̂’s in the respective transformation step. Furthermore, we write 0 instead of 10.
Simplex and Polygon Equations 35
P3= L̂2̂3,178L̂2̂5,279L̂4̂5,470L̂2̂7,389L̂4̂7,580L̂6̂7,690
T7̂,1
= L̂2̂3,178L̂2̂5,279L̂4̂5,470L̂5̂7,690L̂3̂7,589L̂1̂7,356
T5̂,3
= L̂2̂3,178L̂5̂6,690L̂3̂5,479L̂1̂5,246L̂3̂7,589L̂1̂7,356
P2P5= L̂5̂6,690L̂2̂3,178L̂3̂5,479L̂3̂7,589L̂1̂5,246L̂1̂7,356
T3̂,3
= L̂5̂6,690L̂3̂6,589L̂3̂4,478L̂1̂3,145L̂1̂5,246L̂1̂7,356
T1̂,1
= L̂5̂6,690L̂3̂6,589L̂3̂4,478L̂1̂6,356L̂1̂4,245L̂1̂2,123,
which is L̂ᾱ = L̂ω̄ ◦ TC̃o , and
L̂ᾱ = L̂2̂3,178L̂2̂5,279L̂2̂7,389L̂4̂5,470L̂4̂7,580L̂6̂7,690
T2̂,4
= L̂2̂6,389L̂2̂4,278L̂1̂2,123L̂4̂5,470L̂4̂7,580L̂6̂7,690
P2P3= L̂2̂6,389L̂2̂4,278L̂4̂5,470L̂4̂7,580L̂1̂2,123L̂6̂7,690
T4̂,3
= L̂2̂6,389L̂4̂6,580L̂3̂4,478L̂1̂4,245L̂1̂2,123L̂6̂7,690
P3P2P1= L̂2̂6,389L̂4̂6,580L̂6̂7,690L̂3̂4,478L̂1̂4,245L̂1̂2,123
T6̂,4
= L̂5̂6,690L̂3̂6,589L̂1̂6,356L̂3̂4,478L̂1̂4,245L̂1̂2,123
P3= L̂5̂6,690L̂3̂6,589L̂3̂4,478L̂1̂6,356L̂1̂4,245L̂1̂2,123,
which is L̂ᾱ = L̂ω̄ ◦ TC̃e . Hence
TC̃o = T1̂,1T3̂,3P2P5T5̂,3T7̂,1P3 = P3T6̂,4P3P2P1T4̂,3P2P3T2̂,4 = TC̃e ,
which is the heptagon equation.
Example 4.4. Let N = 5. Then ᾱ = (5̂6, 3̂6, 3̂4, 1̂6, 1̂4, 1̂2) ∈ A(b)(6, 4) and η̄ = (4̂56, 2̂56, 2̂36,
2̂34) ∈ A(b)(6, 3). We thus have the chain
4̂56
2̂56
2̂36
2̂34
5̂6−→
12
1̂56
3̂56
2̂36
2̂34
3̂6−→
23
1̂56
1̂36
3̂46
2̂34
3̂4−→
34
1̂56
1̂36
1̂34
3̂45
1̂6−→
12
1̂26
1̂46
1̂34
3̂45
1̂4−→
23
1̂26
1̂24
1̂45
3̂45
1̂2−→
12
1̂23
1̂25
1̂45
3̂45
from which we can read off the position indices in the expression
Lᾱ = L1̂2,1L1̂4,2L1̂6,1L3̂4,3L3̂6,2L5̂6,1.
The Lax system (4.8) consists of the localized pentagon equations
L
k̂k1,a
L
k̂k3,a+1
L
k̂k5,a
=
(
L
k̂k4,a+1
PaLk̂k2,a+1
)
◦ Tk̂,
where k ∈ [6], k̂ = {k1, . . . , k5} with k1 < · · · < k5. The consistency condition is now obtained
as follows. We have
Lᾱ = L1̂2,1L1̂4,2L1̂6,1L3̂4,3L3̂6,2L5̂6,1
P3= L1̂2,1L1̂4,2L3̂4,3L1̂6,1L3̂6,2L5̂6,1
36 A. Dimakis and F. Müller-Hoissen
56
36
34
16
14
12
16
34
26
46
24
45
26
12
45
23
25
23
26
56
25
2446
36
45
12
4534
16
34
12
14
16
56
36
34
16
14
12
13
15
34
36
23
35
56
56
23
56
25
45
23
2356
25
5623
36
45
35
34
1513
16 12
14
Figure 23. Projection of T (7, 4) (Edelman–Reiner polyhedron) to T (6, 3) (associahedron). We use
complementary labeling.
T6̂,1
= L1̂2,1L1̂4,2L3̂4,3L4̂6,2P1L2̂6,2
T4̂,2
= L1̂2,1L4̂5,3P2L2̂4,3P1L2̂6,2
P3= L4̂5,3L1̂2,1P2L2̂4,3P1L2̂6,2 = L4̂5,3L1̂2,1P2P1L2̂4,3L2̂6,2
= L4̂5,3P2P1L1̂2,2L2̂4,3L2̂6,2
T2̂,1
= L4̂5,3P2P1L2̂5,3P2L2̂3,3,
which is Lᾱ = Lω̄ ◦ TC̃o . This yields the left hand side of the hexagon equation (4.4). The right
hand side of (4.4) is obtained if we proceed according to the chain Ce. The full compositions of
maps appearing in this computation can be visualized as maximal chains of T (6, 2), which forms
a 4-hypercube. For larger values of N , the corresponding computation, based on (4.8), becomes
very complicated. This is in contrast to the derivation of the consistency condition using the
hatted version of the Lax system.
Remark 4.5. The derivation of the hexagon equation for the maps Tĵ in Example 4.4 still works
if P is any braiding map (solution of the Yang–Baxter equation), provided that the following
relations hold,
LaLa+1La = La+1PaLa+1, PaPa+1Pa = Pa+1PaPa+1,
LaPa+1Pa = Pa+1PaLa+1, PaPa+1La = La+1PaPa+1,
PaLb = LbPa for |a− b| > 1,
which determine an extension of the braid group. If P is the transposition map, as assumed
in this work outside of this remark, these relations become identities, with the exception of the
first, the pentagon equation.
4.4 Reductions of polygon equations
The relation between the Tamari order T (N,N − 2) and the N -gon equation, together with
the projection of Tamari orders defined in Remark 2.17, induces a relation between neighboring
polygon equations:
T (N,N − 2) ←→ N -gon equation
↓ ↓
T (N − 1, N − 3) ←→ (N − 1)-gon equation
But we have to consider the projection T (N,N − 3) → T (N − 1, N − 4) (for N > 4) in order
to display the full structure of the corresponding polygon equations. Examples are shown in
Figs. 23 and 24. We use the set {0, 1, 2, 3, 4, 5, N − 1} instead of {1, 2, 3, 4, 5, 6, N}. Unlike
the case of simplex equations, there is a substantial difference between odd and even polygon
equations.
Simplex and Polygon Equations 37
12
17
67
14
1537
47
16
13
371557
27
45
34
34
35
45
4527
36
36
2356
25
25
56
56
5623
23
23
67
47
45
27
25
23
27
45
17
37
57
15
35
56
15
37
23
56
13
34
36
12
14
16
1267
14671247
16
34
67144712
45
3416
67
3445
12
27
36
4624
25
56
2626
23
67
47
45
27
25
23
12
24
26
12
45
12
47 14
34
46
12
67
14
67
34
67
16
36
56
16
34
Figure 24. Projection of T (8, 5) to T (7, 4). As in Fig. 23, the coloring marks those parts of the two
Tamari orders that are related by the projection.
Let N be odd, i.e., N = 2n+ 1. Setting
T̂ (2n+1)
ĵ
:= T̂ (2n)
ĵ
, j = 1, . . . , 2n,
and choosing for
T̂ (2n+1)
0̂
: U0̂2 × U0̂4 × · · · × U0̂(2n)
−→ U0̂1 × U0̂3 × · · · U ̂0(2n−1)
the identity map19, it follows that T̂ (2n+1)
ĵ
, j = 0, . . . , 2n, satisfy the (2n+ 1)-gon equation.
Example 4.6. The heptagon equation (with labels 0, 1, . . . , 6) reads
T̂ (7)
0̂,123
T̂ (7)
2̂,145
T̂ (7)
4̂,246
T̂ (7)
6̂,356
= T̂ (7)
5̂,356
T̂ (7)
3̂,245
T̂ (7)
1̂,123
.
If T̂ (7)
0̂
is the identity map, this reduces to the hexagon equation
T̂ (6)
2̂,145
T̂ (6)
4̂,246
T̂ (6)
6̂,356
= T̂ (6)
5̂,356
T̂ (6)
3̂,245
T̂ (6)
1̂,123
.
Let now N be even, i.e., N = 2n. For j ∈ [2n], j 6= 1, we have
T̂ (2n)
ĵ
=
(
u0̂j , T
(2n)
ĵ
Pĵ
)
,
where ĵ = {1, j2, . . . , j2n−1} with 1 < j2 < · · · < j2n−1, and u0̂j ∈ U0̂j . These are maps
T̂ (2n)
ĵ
: U1̂j × Uĵj3 × · · · × Ûjj2n−1
−→ U0̂j × Uĵj2 × Uĵj4 × · · · × Ûjj2n−2
.
Ignoring the first argument of these maps, the 2n-gon equation implies that the resulting maps
have to satisfy the (2n − 1)-gon equation. Conversely, let T̂ (2n−1)
ĵ
, j = 2, 3, . . . , 2n, satisfy the
(2n− 1)-gon equation. We extend these maps trivially via
T̂ (2n)
ĵ
(
u1̂j , uĵj3 , . . . , u ̂jj2n−1
)
:= T̂ (2n−1)
ĵ
(
u
ĵj3
, . . . , u ̂jj2n−1
)
.
Furthermore, we assume that U0̂1 = U1̂2 and U ̂1(2j−1)
= U
1̂(2j)
, j = 2, 3, . . . , n, and we choose
for T̂ (2n)
1̂
the identity map. Then, after dropping the first position index in all terms, we find
that T̂ (2n)
ĵ
, j = 1, . . . , 2n, solve the 2n-gon equation, with hatted indices shifted by 1, and with
position indices shifted by n. In this way, the 2n-gon equation reduces to the (2n − 1)-gon
equation.
19Recall the identifications made in the definition of T̂ (2n).
38 A. Dimakis and F. Müller-Hoissen
Example 4.7. The octagon equation
T̂ (8)
2̂,1,5,6,7
T̂ (8)
4̂,2,5,8,9
T̂ (8)
6̂,3,6,8,10
T̂ (8)
8̂,4,7,9,10
= T̂ (8)
7̂,4,7,9,10
T̂ (8)
5̂,3,6,8,9
T̂ (8)
3̂,2,5,6,7
T̂ (8)
1̂,1,2,3,4
reduces in the way described above to the heptagon equation (with shifted labels)
T̂ (7)
2̂,5,6,7
T̂ (7)
4̂,5,8,9
T̂ (7)
6̂,6,8,10
T̂ (7)
8̂,7,9,10
= T̂ (7)
7̂,7,9,10
T̂ (7)
5̂,6,8,9
T̂ (7)
3̂,5,6,7
.
5 Three color decomposition of simplex equations
The existence of a decomposition of a Bruhat order into a Tamari order, the corresponding dual
Tamari order, and a mixed order, suggests that there should be a way to construct solutions
of a simplex equation from solutions of the respective polygon equation and its dual, provided
a compatibility condition, associated with the mixed order, is fulfilled. As in Section 4, we
associate with K ∈
( [N ]
N−1
)
a map TK and a dual map SK . We set
RK := P ′′(TK × SK)P ′, (5.1)
where P ′, P ′′ are compositions of transposition maps achieving the necessary shuffling of U−→
Po(K)
and U−→
Pe(K)
, respectively of U←−
Po(K)
and U←−
Pe(K)
, so that RK has the correct structure of a simplex
map U−→
P (K)
→ U←−
P (K)
. The (N−1)-simplex equation then indeed reduces to the N -gon equation
for TK and the dual N -gon equation for SK , and an additional compatibility condition. This
includes one of the results in [57]: special solutions of the 4-simplex equation can be constructed
from solutions of the pentagon equation and its dual. The corresponding compatibility condition
is (1.7) in [57].
2-simplex and trigon equation. If N = 3, we consider maps Rij : Ui × Uj → Uj × Ui,
Tij : Ui → Uj , Sij : Uj → Ui. In complementary notation, we have, for example, R3̂ : U2̂3×U1̂3 →
U1̂3 × U2̂3, T3̂ : U2̂3 → U1̂3, and S3̂ : U1̂3 → U2̂3. We set Rk̂ = Tk̂ × Sk̂, hence Rk̂,a = Tk̂,aSk̂,a+1.
The 2-simplex equation R1̂,1R2̂,2R3̂,1 = R3̂,2R2̂,1R1̂,2 then becomes
T1̂,1S1̂,2T2̂,2S2̂,3T3̂,1S3̂,2 = T3̂,2S3̂,3T2̂,1S2̂,2T1̂,2S1̂,3,
which splits into
T1̂T3̂ = T2̂, S2̂ = S3̂S1̂, S1̂T2̂S3̂ = T3̂S2̂T1̂.
The first two are the trigon equation and its dual. The last equation is an additional condition.
A graphical representation of this “decomposition” of the Yang–Baxter equation is shown in
Fig. 25.
3-simplex and tetragon equation. ForN = 4, we haveR1̂ : U1̂4×U1̂3×U1̂2 → U1̂2×U1̂3×U1̂4,
T1̂ : U1̂4×U1̂2 → U1̂3, S1̂ : U1̂3 → U1̂2×U1̂4, etc.20 According to (5.1), we have to setRk̂ = P1(Tk̂×
Sk̂)P2. This means Rk̂(u, v, w) = (Sk̂(v)1, Tk̂(u,w),Sk̂(v)2), where Sk̂(v) =: (Sk̂(v)1,Sk̂(v)2).
The 3-simplex equation (3.5) then leads to
T2̂T4̂,1 = T3̂T1̂,2, S1̂,1S3̂ = S4̂,2S2̂, T1̂,1S2̂,2T3̂,2S4̂,1 = T4̂,2S3̂,1T2̂,1S1̂,2.
The first two equations are the tetragon equation and its dual. The three equations correspond
to B(b)(4, 2), B(r)(4, 2) and B(g)(4, 2), respectively, cf. Example 2.14. Also see Fig. 26.
20In the setting of linear spaces, Tk̂ will be a product and Sk̂ a coproduct.
Simplex and Polygon Equations 39
23
13
12
13
23 12
23
12
13
23
13
12
12
13
12
23
13
23
=
121323 1223= 2312 231312= 13122313 13231213=
Figure 25. Decomposition of the Yang–Baxter equation viewed on the cube B(3, 0) (with complementary
edge labeling, but hats omitted). Following the action of the composition of Yang–Baxter maps on the
left and the right-hand side of the Yang–Baxter equation, we observe that the action splits as indicated
by the three different colors. A set of edges having the same color corresponds to one of three equations,
represented by the graphs in the second row (where all vertices connected by edges having a different color
have been identified). Blue: trigon equation, red: dual trigon equation, green: compatibility condition.
Here, and also in some of the following figures, labels of edges of an initial (final) maximal chain are
marked blue (red), which has no further meaning.
34
24
23
14
13
12
14
23
14
24
34
13
23
34
13
24
12
34
12
23
24
12
13
14
34
24
23
14
13
12
12
13
14
12
23
24
12
34
13
24
13
23
34
14
24
34
14
23
=
23
24
34
1214
2334
13
14 12
=
12
13
23
1434
1223
24
34 14
=
1324
2413
14
12
1324
13 24
23
12
14
3434
23
=
Figure 26. Decomposition of the 3-simplex (tetrahedron, Zamolodchikov) equation, viewed on the
permutahedron formed by B(4, 1). The second row of figures graphically represents the resulting tetragon
equation, dual tetragon equation, and the compatibility condition.
Remark 5.1. By drawing a line through the midpoints of parallel edges, the half-polytopes
of B(4, 1) in Fig. 26 are mapped to the diagrams in Fig. 5, presented in Fig. 27 in a slightly
different way. A tetragon is mapped in this way to a crossing, a hexagon to a node with six legs
and a number k̂ that associates it with the map T̂k̂. The tetragon equation, its dual and the
compatibility equation are then represented by the further graphs in Fig. 27.
4-simplex and pentagon equation. For N = 5, we have R1̂ : U1̂5 × U1̂4 × U1̂3 × U1̂2 →
U1̂2 × U1̂3 × U1̂4 × U1̂5, T1̂ : U1̂5 × U1̂3 → U1̂2 × U1̂4, S1̂ : U1̂4 × U1̂2 → U1̂3 × U1̂5, etc. We have
to set R1̂ = P2T1̂,3S1̂,1P2, etc., hence R̂k̂,1234 = T̂k̂,13Ŝk̂,24P1,2P34 = T̂k̂,13P1,2P34Ŝk̂,13. The
40 A. Dimakis and F. Müller-Hoissen
4
`
3
`
2
`
1
`
1
`
2
`
3
`
4
`
=
4
`
2
`
1
`
3
`
=
3
`
1
`
2
`
4
`
=
4
`
3
`
2
`
1
`
1
`
2
`
3
`
4
`
=
Figure 27. The first line shows the dual of the permutahedron equality in Fig. 26, in the sense of
Remark 5.1. The further diagrams represent the three parts that arise from the “decomposition” of the
3-simplex equation (3.5).
4-simplex equation (3.7) then splits into the pentagon equation (4.3), its dual
Ŝ2̂,12Ŝ4̂,23 = Ŝ5̂,23Ŝ3̂,13Ŝ1̂,12,
and the additional condition
Ŝ1̂,12T̂2̂,13Ŝ3̂,14 T̂4̂,24Ŝ5̂,34 = T̂5̂,24Ŝ4̂,34T̂3̂,14Ŝ2̂,12T̂1̂,13
(cf. (1.7) in [57]). In terms of T and S, we have (4.2) and
S2̂,1P2S4̂,1 = S5̂,2S3̂,1S1̂,2, S1̂,1T2̂,2S3̂,3P1T4̂,2S5̂,1P2 = P2T5̂,3S4̂,2P3T3̂,1S2̂,2T1̂,3.
Also see Fig. 28.
5-simplex and hexagon equation. For N = 6, we have R1̂ : U1̂6 ×U1̂5 ×U1̂4 ×U1̂3 ×U1̂2 →
U1̂2 ×U1̂3 ×U1̂4 ×U1̂5 ×U1̂6, T1̂ : U1̂6 ×U1̂4 ×U1̂2 → U1̂3 ×U1̂5, S1̂ : U1̂5 ×U1̂3 → U1̂2 ×U1̂4 ×U1̂6,
etc. We set Rk̂ = P3P1P2(Tk̂ × Sk̂)P4P2P3. The 5-simplex equation (3.8) then decomposes
into the hexagon equation (4.4) for T , the dual hexagon equation
S1̂,1S3̂,2P3S5̂,1 = P3S6̂,4S4̂,2P1S2̂,2,
and
T1̂,1S2̂,3T3̂,4P6P2P3P2S4̂,4T5̂,3S6̂,1P4P2 = P4P2T6̂,5S5̂,3T4̂,2P5P4P5P1S3̂,2T2̂,3S1̂,5.
See Fig. 29.
6-simplex and heptagon equation. For N = 7, we have R1̂ : U1̂7×U1̂6×U1̂5×U1̂4×U1̂3×
U1̂2 → U1̂2×U1̂3×U1̂4×U1̂5×U1̂6×U1̂7, T1̂ : U1̂7×U1̂5×U1̂3 → U1̂2×U1̂4×U1̂6, S1̂ : U1̂6×U1̂4×U1̂2 →
U1̂3 × U1̂5 × U1̂7, etc. We set Rk̂ = P4P2P3(Tk̂ × Sk̂)P3P2P4 = P4P2P3Tk̂,1Sk̂,4P3P2P4. The
6-simplex equation (3.9) then decomposes into the heptagon equation (4.6), respectively (4.7),
and the dual heptagon equation
S2̂,1P3P4P5S4̂,2P4P3S6̂,1P3 = P3S7̂,4S5̂,2P4P1S3̂,2S1̂,4,
Simplex and Polygon Equations 41
45
35
34
25
24
23
15
14
13
12
25
24
15
34
15
25
35
45
14
14
24
34
45
35
23
13
13
23
34
35
24
14
13
25
45
35
12
34
12
23
24
25
12
13
14
15
45
35
34
25
24
23
15
14
13
12
12
13
14
15
12
23
24
25
34
35
13
13
23
34
35
12
13
45
24
25
14
35
14
24
34
45
15
25
35
45
14
24
15
23
25
34
=
45
25
23
15
35 13
34
12
14
45
25
23
12
24
12
45
14
34
=
34
14
12
24
45 12
45
23
25
34
14
12
13
15
23
35
25
45
=
35
25 14
13
24 15
45
14 25
12
15
15 24
34 23
24
34 35
35 13
24
23 34
24 15
15
12
25 14
45
15
24
13
14 25
35
=
Figure 28. Decomposition of the 4-simplex (Bazhanov–Stroganov) equation, viewed on the Felsner–
Ziegler polyhedron formed by B(5, 2). The resulting pentagon, dual pentagon and compatibility equations
are represented by the graphs in the second row.
56
46
45
36
35
34
26
25
24
23
16
15
14
13
1
36
26
35
16
45
16
26
36
46
56
25
15
15
25
35
45
56
46
34
24
14
14
24
34
45
46
35
36
23
13
13
23
34
35
36
24
25
15
14
13
26
56
46
45
36
35
12
34
12
23
24
25
26
12
13
14
15
16
56
46
45
36
35
34
26
25
24
23
16
15
14
13
12
12
13
14
15
16
12
23
24
25
26
34
35
36
13
13
23
34
35
36
45
46
24
14
14
24
34
45
46
12
13
56
35
25
26
36
15
46
15
25
35
45
56
16
26
36
46
56
14
15
24
16
23
25
35
26
34
36
45
=
12
23
26
56
25
2446
36
45
12
4534
16
34
12
14
16
2356
25
5623
36
45
35
34
1513
16 12
14
=
45 12
15
25
23
14
16
34
36
56
35
13
56
23
45 12
25
12
23
14
45
16
34
34
36
56
24
26
46
16
=
46 13
16
35
26
15
36
25
14
26
35
56
25
36 14
12
25
24
3624
15 1445
15 24
34
26
23
35
46
46 13
13
46 13
35
13 46
1524
26
23
34
36 14 2445
15
24
25
14 36
25
26
35
15
12 14
25
36 56
35
46
16
13
26
=
Figure 29. Decomposition of the 5-simplex equation on the polyhedron formed by B(6, 3). The resulting
hexagon, dual hexagon and compatibility equations are graphically represented in the second row.
respectively,
Ŝ2̂,123Ŝ4̂,245Ŝ6̂,356 = Ŝ7̂,356Ŝ5̂,246Ŝ3̂,145Ŝ1̂,123,
and the compatibility condition
T1̂,1S2̂,3P2P1P3T3̂,5P7P8P4P3S4̂,5P7P4P3P2T5̂,5P7S6̂,3P5T7̂,1P3P4P5P6P2
= P7P5P3P4P5P6P2S7̂,7T6̂,5P7P8P6S5̂,3P5P6P2P1T4̂,3P5P6P7P2S3̂,3T2̂,5S1̂,7,
42 A. Dimakis and F. Müller-Hoissen
6757
47
37
27
46
17
56
17
27
37
47
57
67
36
26
16
16
26
36
46
5667
57
45
35
25
15
15
25
35
45
56
57
46
47
34
24
14
14
24
34
45
46
47
35
36
37
23
13
13
23
34
35
36
37
24
25
26
16
15
14
13
27
12
12
23
24
25
26
27
12 13
14
15
16
17
27
26
25
24
23
17
16
15
14
13 12
12
13
14
15
16
17
12
23
24
25
26
27
34
35
36
37
13
13
23
34
35
36
37
45
46
47
24
14
14
24
34
45
46
47
56
57
35
25
15
15
25
35
45
5657
67
46
36
26
27
37
47
16
57
16
26
36
46
56
67
17
27
37
47
57
67
=
67
47
45
27
25
23
27
45
17
37
57
15
35
56
15
37
23
56
13
34
36
12
14
16
67
47
45
27
25
23
12
24
26
12
45
12
47 14
34
46
12
67
14
67
34
67
16
36
56
16
34
=
56
36
16
67
46
26
16
34
34
34
67
67
67
14
14
12
12
12
12
47
45
24
23
25
27
45
47
56
56
36
34
16
14
12
23
35
37
13
15
17
15
37
57
45
27
23
25
27
45
47
67
=
13
1627
57
15
14
2716
47
37
46
17
12
27
14
2536
47
16
67
1746
37
24
25
14
3625
47
36
461735
26
23
36
1447
25
56
2635
26
17
35
34
45
3526
17
24
37
134624
57
24
15
17
461357
24
15
5713
1357
15
24
571346
17
24
15
57
244613
37
24
17
35
45
34
35
26
35
17
26
56
47
25
14
36
23
26
35
26
1746
47
36
2536
14
25
24
37
46
3625
17
67
16
47
27
14
12
17
4637
27 16
4714
15
57
16 27
13
=
Figure 30. Decomposition of the 6-simplex equation on the polyhedron formed by B(7, 4). The resulting
heptagon, dual heptagon and compatibility equations are represented by the graphs in the last two rows.
respectively
T̂1̂,123Ŝ2̂,145T̂3̂,167Ŝ4̂,268T̂5̂,469Ŝ6̂,379T̂7̂,589 = Ŝ7̂,379T̂6̂,589Ŝ5̂,269T̂4̂,467Ŝ3̂,168T̂2̂,123Ŝ1̂,145.
See Fig. 30.
6 Conclusions
The main result of this work is the existence of an infinite family of “polygon equations” that
generalize the pentagon equation in very much the same way as the simplex equations generalize
the Yang–Baxter equation. The underlying combinatorial structure in case of simplex equations
Simplex and Polygon Equations 43
is given by (higher) Bruhat orders [78, 79, 80]. Underlying the polygon equations are (higher)
Tamari orders.
We also introduced a visualization of simplex as well as polygon equations as deformations
of maximal chains of posets forming 1-skeletons of polyhedra. This geometrical representation
revealed various deep relations between such equations.
An intermediate result, worth to highlight, is the (three color) decomposition in Section 2.2
of any (higher) Bruhat order into a (higher) Tamari order, the corresponding dual Tamari order,
and a “mixed order”. From this we recovered a relation between the pentagon and the 4-simplex
equation, observed in [57]. Moreover, we showed that this is just a special case of a relation
between any simplex equation and its associated polygon equation. Another (more profound)
observation made in [57] concerns a relation between the pentagon equation and the 3-simplex
equation. This seems not to have a corresponding generalization.
Further exploration of the higher polygon equations is required. We expect that they will
play a role in similarly diverse problems as the pentagon equation does. A major task will be
the search for relevant solutions in suitable frameworks. Such a framework could be the KP
hierarchy, since a subclass of its soliton solutions realizes higher Tamari orders [24, 25].
Appendix A: A different view of simplex and polygon equations
Let B be a monoid and N > 2 an integer. With each J ∈
( [N ]
N−2
)
, we associate an invertible
element LJ of B, and with each K ∈
( [N ]
N−1
)
, we associate elements RK , R′K . They shall be
subject to the conditions ([25], also see [20] for a related structure)
LJLJ ′ = LJ ′LJ if E(J) ∩ E(J ′) = ∅, (A.1)
LJRK = RKLJ , LJR
′
K = R′KLJ if J /∈ P (K), (A.2)
L−→
P (K)
R′K = RKL←−P (K)
, (A.3)
where
L−→
P (K)
= LK\{kN−1} · · ·LK\{k1}, L←−
P (K)
= LK\{k1} · · ·LK\{kN−1},
and K = {k1, . . . , kN−1} with k1 < k2 < · · · < kN−1. For any sequence ρ = (J1, . . . , Jr), let
Lρ = LJ1 · · ·LJr . If ρ ∈ A(N,N − 2), then Lρ′ = Lρ for any ρ′ ∈ [ρ], according to (A.1).
Hence Lρ represents [ρ].
Proposition A.1.
R−→
P ([N ])
= R←−
P ([N ])
⇐⇒ R′−→
P ([N ])
= R′←−
P ([N ])
.
Proof. We follow the two maximal chains (2.1) of B(N,N − 2). Let α be the lexicographi-
cal order on
( [N ]
N−2
)
, and ω the reverse lexicographical order. Let us start with LαR
′−→
P ([N ])
=
LαR
′
N̂
· · ·R′
1̂
and move all LJ , J ∈ P (N̂), to consecutive positions in Lα, using (A.1).
Using (A.2), we commute R′
N̂
to the left until we have the substring L−→
P (N̂)
R′
N̂
. Then we use
(A.3) to replace this by RN̂ L←−P (N̂)
. Using (A.2) again, we commute RN̂ to the left of all L’s, thus
obtaining RN̂Lρ1R
′
N̂−1
· · ·R′
1̂
. Continuing in this way, we finally get LαR
′−→
P ([N ])
= R−→
P ([N ])
Lω.
For the second maximal chain of B(N,N − 2), we obtain LαR
′←−
P ([N ])
= R←−
P ([N ])
Lω correspon-
dingly. Since Lα and Lω are invertible (since we assume that the L’s are invertible), the statement
of the proposition follows. �
44 A. Dimakis and F. Müller-Hoissen
The proposition says that the elements RK , K ∈
( [N ]
N−1
)
, satisfy (the algebraic version of) the
(N − 1)-simplex equation if and only if this is so for R′K , K ∈
( [N ]
N−1
)
. Choosing for all R′K the
identity element of B, (A.3) becomes the Lax system L−→
P (K)
= RKL←−P (K)
.
Example A.2. For N = 3, the Lax system reads LiLj = Rij LjLi, 1 ≤ i < j ≤ 3, so that
Rij = LiLjL
−1
i L−1
j =: [Li, Lj ] is a commutator in a group. The condition (A.1) is empty
and (A.2) requires [[Li, Lj ], Lk] = e for i < j, k 6= i, j, where e is the identity element. Hence,
if G is the group 〈g1, g2, g3 | [[g1, g2], g3] = [[g2, g3], g1] = [[g1, g3], g2] = e〉, then Proposition A.1
implies that Rij := [gi, gj ], i < j, satisfy the Yang–Baxter equation. If G is abelian, then
Rij = e.
We are led to the following by the three color decomposition. Let us keep (A.1), but replace
(A.2) and (A.3) by
LJTK = TKLJ , LJT
′
K = T ′KLJ if J /∈ P (K), (A.4)
L−→
Po(K)
T ′K = TKL←−Pe(K)
.
Then we have
T−→
Po([N ])
= T←−
Pe([N ])
⇐⇒ T ′−→
Po([N ])
= T ′←−
Pe([N ])
.
The proof is analogous to that of Proposition A.1, but here we start with Lα(b)T ′−→
Po([N ])
. Choosing
for T ′K the identity element, we have the Lax system L−→
Po(K)
= TKL←−Pe(K)
for (an algebraic version
of) the N -gon equation.
Let us now keep (A.1), but replace (A.2) and (A.3) by
LJSK = SKLJ , LJS
′
K = S′KLJ if J /∈ P (K), (A.5)
L−→
Pe(K)
S′K = SKL←−Po(K)
.
Then we have
S−→
Pe([N ])
= S←−
Po([N ])
⇐⇒ S′−→
Pe([N ])
= S′←−
Po([N ])
.
Here the proof starts with Lα(r)S′−→
Pe([N ])
. Choosing for T ′K the identity element, we have the Lax
system L−→
Pe(K)
= SKL←−Po(K)
for (an algebraic version of) the dual N -gon equation.
Next, let (A.1), (A.4) and (A.5) hold, and in addition
L−→
Pe(K)
T ′K = TKL←−Po(K)
, L−→
Po(K)
S′K = SKL←−Pe(K)
.
For odd N , the mixed equation reads
SN̂TN̂−1
· · ·T2̂S1̂ = T1̂S2̂ · · ·SN̂−1
TN̂ ,
while for even N is has the form
SN̂TN̂−1
· · ·S2̂T1̂ = S1̂T2̂ · · ·SN̂−1
TN̂ .
We find that the mixed equation holds for SK , TK if and only if it holds for S′K , T ′K . The proof
starts with Lα(g)S′
N̂
T ′
N̂−1
· · ·T ′
2̂
S′
1̂
for odd N , and correspondingly for even N .
Simplex and Polygon Equations 45
Remark A.3. In the present framework, the pentagon equation reads21
T1,2,3,4T1,2,4,5T2,3,4,5 = T1,3,4,5T1,2,3,5.
Here we inserted commas, which we mostly omitted before. We translate the labels as follows.
If a label i1, i2, i3, i4 contains a pair with ir+1 = ir + 2 (higher shifts do not appear), then we
make the substitution ir, ir+1 7→ ir(ir + 1), ir+1, where ir (ir + 1) is understood as a two-digit
expression. Finally we drop the very last index of the resulting new label. If there is no index
pair of the above kind in a label, we keep the label, but also drop the very last index. This
translates the above pentagon equation to
T1,2,3T1,23,4T2,3,4 = T12,3,4T1,2,34.
In this form the pentagon equation shows up in Drinfeld’s theory of associators (see, e.g., [4, 36]).
In the same way, the associated (tetragon) Lax equation L1,2,3L1,3,4T
′
1,2,3,4 = L2,3,4L1,2,4 becomes
L1,2L12,3T
′
1,2,3 = L2,3L1,23,
which becomes the twist equation in the context of associators (see, e.g., equation (2) in [4]).
Furthermore, the 3-simplex equation in the present framework is R1,2,3R1,2,4R1,3,4R2,3,4 =
R2,3,4R1,3,4R1,2,4R1,2,3, which translates to
R1,2R1,23R12,3R2,3 = R2,3R12,3R1,23R1,2,
and the hexagon equation T1,2,3,4,5T1,2,3,5,6T1,3,4,5,6 = T2,3,4,5,6T1,2,4,5,6T1,2,3,4,6 becomes
T1,2,3,4T1,2,34,5T12,3,4,5 = T2,3,4,5T1,23,4,5T1,2,3,45,
and so forth.
Acknowledgments
We have to thank an anonymous referee for comments that led to some corrections in our
previous version of Section 2.2.
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1 Introduction
2 Higher Bruhat and Tamari orders
2.1 Higher Bruhat orders
2.2 Three color decomposition of higher Bruhat orders
3 Simplex equations
3.1 Resolutions of B(N+1,N-1) and polyhedra
3.2 Simplex equations and associated polyhedra
3.3 Lax systems for simplex equations
3.4 Reductions of simplex equations
4 Polygon equations
4.1 Resolutions of T(N,N-2) and polyhedra
4.2 Polygon equations and associated polyhedra
4.3 Lax systems for polygon equations
4.4 Reductions of polygon equations
5 Three color decomposition of simplex equations
6 Conclusions
Appendix A: A different view of simplex and polygon equations
References
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