On representation type of a pair of posets with involution
In this paper we consider the problem on classifying the representations of a pair of posets with involution. We prove that if one of these is a chain of length at least 4 with trivial involution and the other is with nontrivial one, then the pair is tame ⇔ it is of finite type ⇔ the poset with n...
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irk-123456789-1566082019-06-19T01:29:20Z On representation type of a pair of posets with involution Bondarenko, V.M. In this paper we consider the problem on classifying the representations of a pair of posets with involution. We prove that if one of these is a chain of length at least 4 with trivial involution and the other is with nontrivial one, then the pair is tame ⇔ it is of finite type ⇔ the poset with nontrivial involution is a ∗-semichain (∗ being the involution). The case that each of the posets with involution is not a chain with trivial one was considered by the author earlier. In proving our result we do not use the known technically difficult results on representation type of posets with involution. 2005 Article On representation type of a pair of posets with involution / V.M. Bondarenko // Algebra and Discrete Mathematics. — 2005. — Vol. 4, № 1. — С. 1–7. — Бібліогр.: 12 назв. — англ. 1726-3255 2000 Mathematics Subject Classification: 15A, 16G. http://dspace.nbuv.gov.ua/handle/123456789/156608 en Algebra and Discrete Mathematics Інститут прикладної математики і механіки НАН України |
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In this paper we consider the problem on classifying the representations of a pair of posets with involution. We
prove that if one of these is a chain of length at least 4 with trivial
involution and the other is with nontrivial one, then the pair is
tame ⇔ it is of finite type ⇔ the poset with nontrivial involution
is a ∗-semichain (∗ being the involution). The case that each of the
posets with involution is not a chain with trivial one was considered by the author earlier. In proving our result we do not use the
known technically difficult results on representation type of posets
with involution. |
| format |
Article |
| author |
Bondarenko, V.M. |
| spellingShingle |
Bondarenko, V.M. On representation type of a pair of posets with involution Algebra and Discrete Mathematics |
| author_facet |
Bondarenko, V.M. |
| author_sort |
Bondarenko, V.M. |
| title |
On representation type of a pair of posets with involution |
| title_short |
On representation type of a pair of posets with involution |
| title_full |
On representation type of a pair of posets with involution |
| title_fullStr |
On representation type of a pair of posets with involution |
| title_full_unstemmed |
On representation type of a pair of posets with involution |
| title_sort |
on representation type of a pair of posets with involution |
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Інститут прикладної математики і механіки НАН України |
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2005 |
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http://dspace.nbuv.gov.ua/handle/123456789/156608 |
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On representation type of a pair of posets with involution / V.M. Bondarenko // Algebra and Discrete Mathematics. — 2005. — Vol. 4, № 1. — С. 1–7. — Бібліогр.: 12 назв. — англ. |
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Algebra and Discrete Mathematics |
| work_keys_str_mv |
AT bondarenkovm onrepresentationtypeofapairofposetswithinvolution |
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2025-07-14T09:00:07Z |
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Algebra and Discrete Mathematics RESEARCH ARTICLE
Number 1. (2005). pp. 1 – 7
c© Journal “Algebra and Discrete Mathematics”
On representation type of a pair of posets with
involution
Vitalij M. Bondarenko
Communicated by V. V. Kirichenko
Dedicated to Yu.A. Drozd on the occasion of his 60th birthday
Abstract. In this paper we consider the problem on classi-
fying the representations of a pair of posets with involution. We
prove that if one of these is a chain of length at least 4 with trivial
involution and the other is with nontrivial one, then the pair is
tame ⇔ it is of finite type ⇔ the poset with nontrivial involution
is a ∗-semichain (∗ being the involution). The case that each of the
posets with involution is not a chain with trivial one was consid-
ered by the author earlier. In proving our result we do not use the
known technically difficult results on representation type of posets
with involution.
Our paper is devoted to study representations of pairs of posets with
involution.
Representations of a pair of posets (without involution) were intro-
duced, in the language of matrices, in [1]. In the same paper, all pairs
of finite type were described. For tame type it did in [2, 3]. Representa-
tions of a pair of posets with involution (in fact, a more general situation)
were studied in [4, Section 5]; as a consequence of Theorem 6, one has a
criterion for such a pair to be tame in the case when each of the posets
with involution is not a chain with trivial one. The case, when one of
these is a chain with trivial involution, easily reduces to representations
of posets with involution (see Remark at the end of this paper), and we
can use the results from [5, 6] on classifying the tame and wild posets
with involution. All statements of this paper are proved without using
these, technically difficult, results of [5, 6].
2000 Mathematics Subject Classification: 15A, 16G.
Key words and phrases: semichain, tame, wild, representation, category.
2On representation type of a pair of posets with involution
1. Formulation of the main result
Throughout the paper, K denotes a fixed field. All posets are finite and all
K-vector spaces are finite-dimensional. Singletons are always identified
with the elements themselves. The category of (finite-dimensional) K-
vector spaces is denoted as usual by mod K. For simplicity, an involution
on a poset is denoted by the same symbol ∗.
We stress that we keep the right-side notation (vector spaces are right;
linear maps are wrote to the right of elements of spaces and are composed
from left to right, etc.).
Before formulation of our main result we introduce some definitions
and notation (see [4, 7]).
Let X = (A, ∗) be a poset with involution. By an X-graded vector
space over K we mean the direct sum U = ⊕a∈AUa of K-vector spaces Ua
such that Ua∗ = Ua for each a ∈ A. For B ⊆ A, put UB = ⊕y∈BUy; for
B, C ⊆ A and a linear map ϕ : U → U ′ with U and U ′ being X-graded
spaces, denote by ϕBC the map of UB into U ′
C induced by ϕ. A linear
map ϕ : U → U ′ between the X-graded spaces U and U ′ is said to be
an X-map if ϕa∗a∗ = ϕaa for any a and ϕbc = 0 for any b 6≤ c. The
set of X-maps of U into U ′ is denoted by HomX,K(U, U ′). The category
of X-graded vector spaces over K (which has as objects the X-graded
spaces and as morphisms the X-maps) is denoted by modXK.
Since it is natural to identify a poset A with trivial involution ∗ (when
a∗ = a for any a ∈ A) with the poset itself, the above definitions involve
the case of usual posets.
Let S = (A, ∗) and T = (B, ∗) be posets with involution and set
M = modSK, N = modT K. A representation of the pair (S, T ) is a
triple (V, U, λ) formed by objects V ∈ M , U ∈ N and a linear map λ
of V into U . A morphism from (V, U, λ) to (V ′, U ′, λ′) is determined by
morphisms µ : V → V ′ and ν : U → U ′ such that λν = µλ′. The category
defined in this way is denoted by repK(S, T ).
We say that a pair (S, T ) is of finite type if repK(S, T ) has finitely
many isomorphism classes of indecomposables, and tame or wild if so is
the problem of classifying, up to isomorphism, the objects of repK(S, T ).
For the precise definitions of tame and wild pairs see [4, Section 5]; these
are particular cases of Drozd’s definitions [8, 9] (the definition of wild
pairs will be recalled in Subsection 2.1).
A semichain of length m is by definition a poset of the form X =⋃m
i=1 Xi, where each Xi (called a link of X) consist of either one point
or two incomparable points, and x1 < x2 < · · · < xm for any x1 ∈
X1, x2 ∈ X2, . . . , xm ∈ Xm (if every Xi is a singleton, the poset X is
called a chain). A poset with involution T = (Y, ∗) is called a semichain
V. M. Bondarenko 3
(respectively, chain) if Y is a semichain (respectively, chain) and ∗ is
trivial, and a ∗-semichain if Y is a semichain and x∗ = x for each x from
the union of all two-point links (to say that T is a ∗-chain means that Y
is a chain).
Under the assumption that K is algebraic closed, we prove the fol-
lowing theorem.
Theorem. Let S and T be posets with involution such that the first
(respectively, second) one is a chain of length at least 4. Then the follow-
ing conditions are equivalent:
1) (S, T ) is tame;
2) (S, T ) is of finite type;
3) S (respectively, T ) is a ∗-semichain.
This theorem will be proved in Subsections 2.2 and 2.3.
2. Proof of Theorem
2.1. The definition of wild pairs. In this part of the paper we recall
the definition of a wild pair (S, T ) (S, T being posets with involution).
Let X = (A, ∗) be a poset with involution and K̂ = K〈x, y〉 be the
free (associative) K-algebras in two noncommuting variables x and y.
We can define the category modX K̂ of (right) free X-graded K̂-modules
similar to the category modXK, considering finitely generated free K̂-
modules instead of finite-dimensional K-vector spaces (such a module
U = ⊕a∈AUa is called free if so are all the modules Ua, a ∈ A).
Let S = (A, ∗) and T = (B, ∗) be posets with involution, and let
R
K̂
(S, T ) denotes the set of all triples (V, U, λ) formed by objects V ∈
modS K̂, U ∈ modT K̂ and a homomorphism λ from V into U . By L(K̂)
we denote the category of left finite-dimensional (over K) K̂-modules.
The pair (S, T ) is called wild if there exists an element M = (V, U, λ) of
R
K̂
(S, T ) such that the functor
H(M) = M ⊗− : L(K̂) → repK(S, T )
preserves indecomposability and isomorphism classes (M ⊗ Y = (V ⊗
Y, λ ⊗ 1Y , V ⊗ Y ) with (V ⊗ Y )a = Va ⊗ Y and (U ⊗ Y )a = Ua ⊗ Y
for every a); here all tensor products are considered over K̂. An element
M ∈ R
K̂
(S, T ) with such properties will be called strict.
For an element M = (V, U, λ) of R
K̂
(S, T ) with V = K̂n, Va = K̂na ,
U = K̂m and Ub = K̂mb (a ∈ A, b ∈ B), we identify the map λ with
4On representation type of a pair of posets with involution
the matrix (λab), a ∈ A, b ∈ B, where λab is the map of Va into Ub
induced by λ; in turn, we identify each λab with the corresponding na×mb
matrix with entries in K̂. So λ is identified with the block matrix (λab),
a ∈ A, b ∈ B, of order n × m. We will indicate strict elements in this
form.
2.2. Subsidiary lemma. When we determine a poset with involu-
tion S = (A, ∗), the order relation is given up to transitivity and a∗ is
indicated only if a∗ 6= a. If the elements of A are natural numbers, the
order relation is denoted by ≺ (to distinguish between the given relation
and the natural ordering < of the integer numbers); the only excep-
tion is the case when A is linear ordered. By 〈m〉 we denote the chain
{1 < 2 < . . . < m}.
Lemma. Let S = 〈m〉 and T = (B, ∗) be a poset with nontrivial involu-
tion. Then the pair (S, T ) is wild in the following cases:
(a) m = 2, B = {1, 2}, 1∗ = 2;
(b) m = 3, B = {1, 2, 3, 4 | 1 ≺ 3, 2 ≺ 3 ≺ 4}, 1∗ = 3, 2∗ = 4;
(c) m = 3, B = {1, 2, 3, 4 | 1 ≺ 2 ≺ 3, 2 ≺ 4}, 1∗ = 3, 2∗ = 4;
(d) m = 3, B = {1, 2, 3, 4 | 1 ≺ 2 ≺ 4, 1 ≺ 3 ≺ 4}, 1∗ = 3, 2∗ = 4;
(e) m = 4, B = {1, 2, 3 | 1 ≺ 3, 2 ≺ 3}, 1∗ = 3;
(f) m = 4, B = {1, 2, 3 | 1 ≺ 2, 1 ≺ 3}, 1∗ = 3.
Proof. We will use the definitions, notation and conventions of 2.1. Writ-
ing a strict element M = (V, U, λ) of R
K̂
(S, T ), we assume that rth hor-
izontal (respectively, vertical) band of λ is situated above (respectively,
to the left of) sth one if r < s.
As a strict element L = (V, U, λ) of R
K̂
(S, T ) one can take the fol-
lowing one:
in case (a), V = K̂2, V1 = V2 = K̂, U = K̂4, U1 = U2 = K̂2, and
λ =
(
1 0 0 x
0 1 1 y
)
;
in case (b), V = K̂10, V1 = K̂2, V2 = K̂3, V3 = K̂5, U = K̂16, U1 =
V. M. Bondarenko 5
U2 = U3 = U4 = K̂4, and
λ =
0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0
0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0
0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0
1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0
0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 x
0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 y
;
in cases (c) and (d), V, U and λ are the same as these in case (b),
but one must arrange the vertical bands λ1, λ2, λ3, λ4 of λ as follows:
λ4, λ3, λ2, λ1 and λ3, λ2, λ1, λ4, respectively;
in case (e), V = K̂12, V1 = K̂2, V2 = V3 = K̂3, V4 = K̂4, U =
K̂16, U1 = K̂6, U2 = K̂4, U3 = K̂6, and
λ =
0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0
1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 x
0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0
0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 y
0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0
0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 1 0 1 0 1 0
0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 1
;
in case (f), V, U and λ are the same as these in case (e), but one must
arrange the vertical bands λ1, λ2, λ3 of λ in reverse order.
The fact that each of these elements of R
K̂
(S, T ) is strict can be
proved in the same way as in the case when the poset with involution S
is not a chain (see [4, Section 5]).
2.3. Completion of the proof. We proceed now immediately to
the proof of the theorem. By Proposition 7 of [4] (S, T ) is tame if and
6On representation type of a pair of posets with involution
only if so is (T op, Sop), and hence one may assume that S is a chain of
length at least 4 (Sop, T op denote the dual posets with involution).
The implication 2) ⇒ 1) is trivial. The implication 3) ⇒ 2) fol-
lows from the fact that, for any ∗-semichain T = (B, ∗), the category
repK(S, T ) is isomorphic to the category of representations of a bundle of
the chain S and the semichain B (with the involution on S ∪ B induced
by ∗); this bundle is of finite type by the main classification theorem of
[10, §1]. Finally, in view of the fact that any (S, T ) is either tame or wild
(see the main result of [9]), and Lemma 8 of [4], the implication 1) ⇒ 3) is
equivalent to the assertion that (S, T ) with S = 〈4〉 is wild if T = (B, ∗)
is not a ∗-semichain. This assertion immediately follows from Lemma 12
of [4] and the above lemma.
2.4. Remark. One can easily show (in an analogous way as in [1,
§2] for representations of pairs of posets without involution) that in the
case S = 〈m〉, T = (B, ∗) with m > 1 the problem of classifying, up
to isomorphism, the objects of R = rep(S, T ) reduces to that for the
category R′ of representations of the poset with involution T (m − 1) =
〈m − 1〉
∐
T (here
∐
denotes the direct sum of posets with involution),
and we can apply the main results of the theory of representations of
posets with involution [5, 6] (see also [11]).
All statements of this paper have proved without using the (techni-
cally difficult) results of [5, 6, 11]. For readers interested in all points of
view to our theory, we give here a sketch of a proof of Lemma (respec-
tively, of the implication 3) ⇒ 1) of Theorem) with using ones.
In cases (b)–(f) of Lemma, using the algorithm from [6, §4], one can
reduce the problem of classifying the representations of T (m− 1) to that
of some poset with trivial involution T ′(m−1). Since T ′(m−1) contains
(in each of cases (b)–(f)) a subposet isomorphic to 〈1〉
∐
〈3〉
∐
〈4〉, it is
wild by [12]; then T (m − 1) is also wild and consequently so is the pair
(S, T ). In case (a) the pair (S, T ) is wild, for example, by Main theorem
of [6].
As to Theorem, the implication 3) ⇒ 1) follows from the fact that,
for any ∗-semichain T = (B, ∗), the poset T ′(m − 1) is of finite type.
References
[1] M. M. Kleiner, Partially ordered sets of finite type, Zap. Nauchn. Sem. Leningrad.
Otdel. Mat. Inst. Steklov. (LOMI) 28 (1972), 32-41 (in Russian).
[2] M. M. Kleiner, Pairs of partially ordered sets of tame representation type, Linear
Algebra Appl., 104 (1988), 103-115.
[3] A. S. Shkabara, Quivers with relations and DGC, Akad. Nauk Ukrain. SSR, Inst.
Mat., Preprint No. 78.43 (1978), 3-41 (in Russian).
[4] V. M. Bondarenko, Linear operators on S-graded vector spaces, Linear Algebra
Appl., 365 (2003), 45-90.
V. M. Bondarenko 7
[5] V. M. Bondarenko, A. G. Zavadskij, Posets with an equivalence relation of tame
type and of finite growth, Canad. Math. Soc. Conf. Proc., 11 (1991), 67-88.
[6] V. M. Bondarenko, A. G. Zavadskij, Tame posets with equivalence relation, Con-
tem. Math., 131, part 2 (1992), 237-251.
[7] V. M. Bondarenko, On dispersing representations of quivers and their connection
with representations of bundles of semichains, Algebra and discrete math., 1
(2002), 19-31.
[8] Y. A. Drozd, On tame and wild matrix problems, Matrix problems, Akad. Nauk
Ukrain. SSR, Inst. Mat., Kiev, 1977, 104–114 (in Russian).
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Forms, Akad. Nauk Ukrain. SSR, Inst. Mat., Kiev, 1979, 39-74 (English transl.
in AMS Translations, 128 (1986), 31-55).
[10] V. M. Bondarenko, Representations of bundles of semichained sets, and their
applications, Algebra i Analiz, 3 (1991), 38-61 (English transl. in St. Petersburg
Math. J., 3 (1992) 973-996).
[11] L. A. Nazarova, A. V. Roiter, Representations and forms of weakly completed
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Nauk Ukrain. SSR, Inst. Mat., Kiev, 1983, 19–54 (in Russian).
[12] L. A. Nazarova, Partially ordered sets of infinite type, Izv. Akad. Nauk SSSR, 39
(1975), 963-991 (in Russian).
Contact information
V.M. Bondarenko Institute of Mathematics,
Tereshchenkivska 3, 01601 Kyiv, Ukraine
E-Mail: vit-bond@imath.kiev.ua
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