String Functions for Affine Lie Algebras Integrable Modules
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irk-123456789-1480762019-02-17T01:26:09Z String Functions for Affine Lie Algebras Integrable Modules Kulish, P. Lyakhovsky, V. 2008 Article String Functions for Affine Lie Algebras Integrable Modules / P. Kulish, V. Lyakhovsky // Symmetry, Integrability and Geometry: Methods and Applications. — 2008. — Т. 4. — Бібліогр.: 14 назв. — англ. 1815-0659 2000 Mathematics Subject Classification: 17B10; 17B67 http://dspace.nbuv.gov.ua/handle/123456789/148076 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України |
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Kulish, P. Lyakhovsky, V. String Functions for Affine Lie Algebras Integrable Modules Symmetry, Integrability and Geometry: Methods and Applications |
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Kulish, P. Lyakhovsky, V. |
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String Functions for Affine Lie Algebras Integrable Modules |
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String Functions for Affine Lie Algebras Integrable Modules |
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String Functions for Affine Lie Algebras Integrable Modules |
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String Functions for Affine Lie Algebras Integrable Modules |
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String Functions for Affine Lie Algebras Integrable Modules |
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string functions for affine lie algebras integrable modules |
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Інститут математики НАН України |
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String Functions for Affine Lie Algebras Integrable Modules / P. Kulish, V. Lyakhovsky // Symmetry, Integrability and Geometry: Methods and Applications. — 2008. — Т. 4. — Бібліогр.: 14 назв. — англ. |
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Symmetry, Integrability and Geometry: Methods and Applications |
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 4 (2008), 085, 18 pages
String Functions
for Affine Lie Algebras Integrable Modules?
Petr KULISH † and Vladimir LYAKHOVSKY ‡
† Sankt-Petersburg Department of Steklov Institute of Mathematics,
Fontanka 27, 191023, Sankt-Petersburg, Russia
E-mail: kulish@euclid.pdmi.ras.ru
‡ Department of Theoretical Physics, Sankt-Petersburg State University,
1 Ulyanovskaya Str., Petergof, 198904, Sankt-Petersburg, Russia
E-mail: Vladimir.Lyakhovsky@pobox.spbu.ru
Received September 15, 2008, in final form December 04, 2008; Published online December 12, 2008
Original article is available at http://www.emis.de/journals/SIGMA/2008/085/
Abstract. The recursion relations of branching coefficients k(µ)
ξ for a module Lµ
g↓h reduced
to a Cartan subalgebra h are transformed in order to place the recursion shifts γ ∈ Γa⊂h
into the fundamental Weyl chamber. The new ensembles FΨ (the “folded fans”) of shifts
were constructed and the corresponding recursion properties for the weights belonging to the
fundamental Weyl chamber were formulated. Being considered simultaneously for the set
of string functions (corresponding to the same congruence class Ξv of modules) the system
of recursion relations constitute an equation MΞv
(u)m
µ
(u) = δµ
(u) where the operator MΞv
(u) is
an invertible matrix whose elements are defined by the coordinates and multiplicities of
the shift weights in the folded fans FΨ and the components of the vector mµ
(u) are the
string function coefficients for Lµ enlisted up to an arbitrary fixed grade u. The examples
are presented where the string functions for modules of g = A
(1)
2 are explicitly constructed
demonstrating that the set of folded fans provides a compact and effective tool to study the
integrable highest weight modules.
Key words: affine Lie algebras; integrable modules; string functions
2000 Mathematics Subject Classification: 17B10; 17B67
1 Introduction
We consider integrable modules Lµ with the highest weight µ for affine Lie algebra g and
are especially interested in the properties of the string functions related to Lµ. String func-
tions and branching coefficients of the affine Lie algebras arise in the computation of the local
state probabilities for solvable models on square lattice [1]. Irreducible highest weight modules
with dominant integral weights appear also in application of the quantum inverse scattering
method [2] where solvable spin chains are studied in the framework of the AdS/CFT correspon-
dence conjecture of the super-string theory (see [3, 4] and references therein).
There are different ways to deal with string functions. One can use the BGG resolution [5]
(for Kac–Moody algebras the algorithm is described in [6, 7]), the Schur function series [8], the
BRST cohomology [9], Kac–Peterson formulas [6] or the combinatorial methods applied in [10].
Here we want to develop a new description for string functions by applying the recursive
formulas for weight multiplicities and branching coefficients obtained in [11].
?This paper is a contribution to the Special Issue on Kac–Moody Algebras and Applications. The full collection
is available at http://www.emis.de/journals/SIGMA/Kac-Moody algebras.html
mailto:kulish@euclid.pdmi.ras.ru
mailto:Vladimir.Lyakhovsky@pobox.spbu.ru
http://www.emis.de/journals/SIGMA/2008/085/
http://www.emis.de/journals/SIGMA/Kac-Moody_algebras.html
2 P. Kulish and V. Lyakhovsky
It was proved in [6] that for simply laced or twisted affine Lie algebra and integrable modu-
le Lµ with the highest weight µ of level 1 the string function is unique:
σ
(
e−δ
)
:=
∞∏
n=1
1
(1− e−nδ)mult(nδ)
.
so that the corresponding formal character ch (Lµ) can be easily written down provided the set
max(µ) of maximal weights for Lµ is known:
ch (Lµ) = σ
(
e−δ
) ∑
α∈M
eµ+α−
(
|α|2
2
+(µ|α)
)
δ (1)
with
M :=
r∑
i=1
Zα∨i for untwisted algebras or A(2)
2r
r∑
i=1
Zαi for A(u≥2)
r and A 6= A
(2)
2r
(see also Corollary 2.1.6 in [12]). Comparing this expression with the Weyl–Kac formula
ch (Lµ) =
1
R
∑
w∈W
ε(w)ew◦(µ+ρ)−ρ,
where the character can be treated as generated by the denominator 1
R acting on the set of
singular vectors Ψ(µ) =
∑
w∈W
ε(w)ew◦(µ+ρ)−ρ of the module Lµ we see that in the relation (1)
both factors on the right hand side are simplified: singular weights are substituted by the
maximal ones and instead of the factor 1
R the string function σu
(
e−δ
)
is applied.
In this paper we shall demonstrate that similar transformations can be defined when the
level k (µ) is arbitrary. To find these transformations we use the recursion properties of branching
coefficients k(µ)
ξ for the reduced module Lµg↓a where the subalgebraa has the same rank as g:
r(a) = r(g). These properties are formulated in [11] in terms of relations
k
(µ)
ξ =
∑
γ∈Γa⊂g
s (γ) k(µ)
ξ+γ +
∑
w∈W
ε (w) δξ,πa◦(w◦(µ+ρ)−ρ),
where πa is the projection to the weight space of a and Γa⊂g is the fan of the injection a −→ g,
that is the set of vectors defined by the relation
1−
∏
α∈(πa◦∆+)
(
1− e−α
)mult(α)−multa(α) =
∑
γ∈Γa⊂g
s (γ) e−γ
(with s (γ) 6= 0). In particular when a is a Cartan subalgebra h of g the coefficients k(µ)
ξ are
just the multiplicities of the weights of Lµ and the corresponding fan Γh⊂g coincides with Ψ̂(0) –
the set of singular weights ψ ∈ P for the module L0.
In Section 3 we demonstrate that this set can be “folded” Ψ̂(0) −→ FΨ so that the new
shifts (the vectors of the folded fan) fψ ∈ FΨ connect only the weights in the closure of the
fundamental Weyl chamber while the recursive property survives in a new form. Thus the recur-
sive relations are obtained for the coefficients of the string functions for the modules Lξj whose
highest weights ξj belong to the same congruence class Ξk;v. When these relations are applied
simultaneously to the set of string functions located in the main Weyl chamber (Section 4) this
String Functions for Affine Lie Algebras Integrable Modules 3
results in the system of linear equations for the string function coefficients (collected in the
vectors m(µ)
(s,u)). This system can be written in a compact form MΞv
(u)m
µ
(u) = δµ(u) where the ope-
rator MΞv
(u) is a matrix whose elements are composed by the multiplicities of weights in the folded
fans FΨ. The set is solvable and the solution – the vector mµ
(u) – defines the string functions
for Lµ up to an arbitrary minimal grade u. In the Section 5 some examples are presented where
the string functions for modules of g = A
(1)
2 are explicitly constructed.
The set of folded fans provides a compact and effective method to construct the string
functions.
2 Basic definitions and relations
Consider the affine Lie algebra g with the underlying finite-dimensional subalgebra
◦
g.
The following notation will be used:
Lµ – the integrable module of g with the highest weight µ;
r – the rank of the algebra g;
∆ – the root system;
∆+ – the positive root system for g;
mult (α) – the multiplicity of the root α in ∆;
◦
∆ – the finite root system of the subalgebra
◦
g;
N µ – the weight diagram of Lµ;
W – the corresponding Weyl group;
C(0) – the fundamental Weyl chamber;
ρ – the Weyl vector;
ε (w) := det (w), w ∈W ;
αi – the i-th simple root for g, i = 0, . . . , r;
δ – the imaginary root of g;
α∨i – the simple coroot for g, i = 0, . . . , r;
◦
ξ – the finite (classical) part of the weight ξ ∈ P ;
λ =
(◦
λ; k;n
)
– the decomposition of an affine weight indicating the finite part
◦
λ, level k and
grade n;
C
(0)
k – the intersection of the closure of the fundamental Weyl chamber C(0) with the plane
with fixed level k = const;
P – the weight lattice;
Q – the root lattice;
M :=
r∑
i=1
Zα∨i for untwisted algebras or A(2)
2r ,
r∑
i=1
Zαi for A(u≥2)
r and A 6= A
(2)
2r ,
;
E – the group algebra of the group P ;
Θλ := e−
|λ|2
2k
δ ∑
α∈M
etα◦λ – the classical theta-function;
Aλ :=
∑
s∈
◦
W
ε(s)Θs◦λ;
Ψ(µ) := e
|µ+ρ|2
2k
δ − ρAµ+ρ = e
|µ+ρ|2
2k
δ − ρ ∑
s∈
◦
W
ε(s)Θs◦(µ+ρ) =
=
∑
w∈W
ε(w)ew◦(µ+ρ)−ρ – the singular weight element for the g-module Lµ;
4 P. Kulish and V. Lyakhovsky
Ψ̂(µ) – the set of singular weights ψ ∈ P for the module Lµ with the coordinates( ◦
ψ, k, n, ε (w (ψ))
)
|ψ=w(ψ)◦(µ+ρ)−ρ (this set is similar to P ′
nice (µ) in [7]);
m
(µ)
ξ – the multiplicity of the weight ξ ∈ P in the module Lµ;
ch (Lµ) – the formal character of Lµ;
ch (Lµ) =
∑
w∈W
ε(w)ew◦(µ+ρ)−ρ
∏
α∈∆+
(1−e−α)mult(α) = Ψ(µ)
Ψ(0) – the Weyl–Kac formula;
R :=
∏
α∈∆+
(1− e−α)mult(α) = Ψ(0) – the denominator;
max(µ) – the set of maximal weights of Lµ;
σµξ (q) =
∞∑
n=0
m
(µ)
(ξ−nδ)q
n – the string function through the maximal weight ξ.
3 Folding a fan
The generalized Racah formula for weight multiplicities m(µ)
ξ (with ξ ∈ P ) in integrable highest
weight modules Lµ (g) (see [13] for a finite dimensional variant),
m
(µ)
ξ = −
∑
w∈W\e
ε(w)m(µ)
ξ−(w◦ρ−ρ) +
∑
w∈W
ε(w)δ(w◦(µ+ρ)−ρ),ξ, (2)
can be obtained as a special case of developed in [11] (see also [14]) branching algorithm for
affine Lie algebras. To apply this formula (2) we must determine two sets of singular weights:
Ψ̂(µ) for the module Lµ and Ψ̂(0) for L0. (As it was indicated in the Introduction the set Ψ̂(0)
coincides with the fan Γh⊂g of the injection h −→ g of the Cartan subalgebra h in the Lie
algebra g.)
Our main idea is to contract the set Ψ̂(0) (the fan Γh⊂g) into the closure C(0) of the funda-
mental Weyl chamber C(0). We shall use the set max(µ) of maximal weights of Lµ (g) instead
of Ψ̂(µ). And as a result we shall find the possibility to solve the relations based on the recur-
rence properties of weight multiplicities, to obtain the explicit expressions for the string functions
σµξ∈max(µ) and thus to describe the module Lµ.
Consider the module Lµ (g) of level k: µ =
(◦
µ; k; 0
)
. Let C(0)
k;0 be the intersection of C(0)
k with
the plane δ = 0, that is the “classical” part of the closure of the affine Weyl chamber at level k.
To each ξ ∈ P attribute a representative wξ ∈W of the class of transformations
wξ ∈W/Wξ, Wξ := {w ∈W |w ◦ ξ = ξ} ,
bringing the weight ξ into the chamber C(0)
k{
wξ ◦ ξ ∈ C
(0)
k | ξ ∈ P,wξ ∈W/Wξ
}
.
Fix such representatives for each shifted vector φ (ξ, w) = ξ − (w ◦ ρ− ρ). The set{
wφ(ξ,w) | wφ(ξ,w) ◦ φ (ξ, w) ∈ C(0)
k
}
,
is in one-to-one correspondence with the set {φ (ξ, w)} of shifted weights. The recursion rela-
tion (2) can be written as
m
(µ)
ξ = −
∑
w∈W\e
ε(w)m(µ)
φ(ξ,w) +
∑
w∈W
ε(w)δ(w◦(µ+ρ)−ρ),ξ
String Functions for Affine Lie Algebras Integrable Modules 5
= −
∑
w∈W\e
ε(w)m(µ)
wφ(ξ,w)◦φ(ξ,w) +
∑
w∈W
ε(w)δ(w◦(µ+ρ)−ρ),ξ.
Consider the restriction to C(0)
k :
m
(µ)
ξ
∣∣∣
ξ∈C(0)
k
= −
∑
w∈W\e
ε(w)m(µ)
wφ(ξ,w)◦φ(ξ,w) + δµ,ξ. (3)
In the r.h.s. the function m(µ)
ξ′ has an argument ξ′ = wφ(ξ,w) ◦ φ (ξ, w) ∈ C(0)
k :
m
(µ)
ξ′ = m
(µ)
wφ(ξ,w)◦φ(ξ,w) = m
(µ)
ξ+(wφ(ξ,w)◦φ(ξ,w)−ξ).
Thus the new (“folded”) shifts are introduced:
fψ (ξ, w) :=
(
ξ′ − ξ
)
ξ′ 6=ξ = wφ(ξ,w) ◦ (ξ − (w ◦ ρ− ρ))w 6=e − ξ, ξ, ξ′ ∈ C(0)
k , ξ′ 6= ξ.
When the sum over W \ e in the expression (3) is performed the shifted weight ξ′ acquires the
(finite) multiplicity η̂ (ξ, ξ′):
η̂
(
ξ, ξ′
)
= −
∑
w∈W\e,
ε(w), (4)
(the sum is over all the elements w ∈W \e satisfying the relation wφ(ξ,w̃)◦(ξ − (w ◦ ρ− ρ)) = ξ′)
such that
m
(µ)
ξ
∣∣∣
ξ∈C(0)
k
=
∑
ξ′∈C(0)
k ,ξ′ 6=ξ
η̂
(
ξ, ξ′
)
m
(µ)
ξ+fψ(ξ,w) + δξ,µ. (5)
The main property of the multiplicities η̂ (ξ, ξ′) is that they do not depend directly on nξ.
Lemma 1. Let ψ = ρ − w ◦ ρ; φ (ξ, w) = ξ + ψ; ξ′ := wφ(ξ,w) ◦ φ (ξ, w); ξ, ξ′ ∈ C
(0)
k . Then
the corresponding folded shifts fψ (ξ, w) = ξ′− ξ and multiplicities η̂ (ξ, ξ′) depend only on k,
◦
ξ,
and w.
Proof. As far as imaginary roots are W -stable we have: wφ(ξ,w) ◦ (ξ + ñδ) = wφ(ξ,w) ◦ ξ + ñδ.
Thus for both ξ and ξ̃ = ξ + ñδ the representatives of the classes bringing φ (ξ, w) and φ
(
ξ̃, w
)
to the fundamental chamber C(0)
k can be taken equal: wφ(ξ,w) = w
φ(ξ̃,w) modWξ. In the shift
fψ (ξ, w) decompose the element wφ(ξ,w) = tφ(ξ,w) · sφ(ξ,w) into the product of the classical
reflection sφ(ξ,w) and the translation tφ(ξ,w). Denote by θ∨φ(ξ,w) the argument (belonging to M)
of the translation tφ(ξ,w). The direct computation demonstrates that the weight fψ (ξ, w) does
not depend on nξ:
fψ (ξ, w) =
sφ(ξ,w) ◦
(◦
ξ +
◦
ψ
)
−
◦
ξ + k
◦
θ
∨
φ(ξ,w), 0,
n−w◦ρ − k
2
∣∣θ∨φ(ξ,w)
∣∣2 − (
sφ(ξ,w) ◦
(◦
ξ +
◦
ψ
)
, θ∨φ(ξ,w)
)
.
Thus the shift fψ (ξ, w) can be considered as depending on k,
◦
ξ and w: fψ = fψ
(◦
ξ, k, w
)
. The
multiplicity η̂ (ξ, ξ′) (see (4)) depends only on the set of reflections w ∈ W connecting ξ and
ξ′ 6= ξ and does not depend on nξ neither: η̂ (ξ, ξ′) = η̂
(◦
ξ, k, ξ′
)
. �
6 P. Kulish and V. Lyakhovsky
Thus we have constructed the set of (nonzero) shifts fψ
(◦
ξ, k, w
)
with the multiplicities
η̂
(◦
ξ, k, ξ + fψ
(◦
ξ, k, w
))
and obtained the possibility to formulate the recursion properties en-
tirely defined in the closure C(0)
k of the fundamental Weyl chamber.
Let us return to the relation (5),
m
(µ)
ξ
∣∣∣
ξ∈C(0)
k
=
∑
ξ′∈C(0)
k , ξ′ 6=ξ
η̂
(◦
ξ, k, ξ′
)
m
(µ)
ξ+fψ(
◦
ξ,k,w)
+ δξ,µ
=
∑
fψ(
◦
ξ,k,w) 6=0
η̂
(◦
ξ, k, ξ + fψ
(◦
ξ, k, w
))
m
(µ)
ξ+fψ(
◦
ξ,k,w)
+ δξ,µ.
For simplicity from now on we shall omit some arguments and write down the shifts as fψ
(◦
ξ
)
and their multiplicities as η̂
(◦
ξ, ξ′
)
(keeping in mind that we are at the level k and the weight ξ′
depends on the initial reflection w). The set of vectors:
F̃Ψ
(◦
ξ
)
:=
{
ξ′ − ξ = fψ
(◦
ξ
)
=
( ◦
fψ
(◦
ξ
)
; 0;n
fψ
(◦
ξ
))∣∣ξ′ − ξ 6= 0
}
,
ξ′ = wφ(ξ,w) ◦ φ (ξ, w) , ξ, ξ′ ∈ C(0)
k ,
plays here the role similar to that of the set
{
Ψ(0) \ 0
}
of nontrivial singular weights for L0 in
the relation (2) and is called the folded fan for
◦
ξ. (The initial (unfolded) fan Γh⊂g corresponds
here to the injection of the Cartan subalgebra.)
Thus we have proved the following property:
Proposition 1. Let Lµ be the integrable highest weight module of g, µ =
(◦
µ; k; 0
)
, ξ =(◦
ξ; k;nξ
)
∈ N µ, ξ ∈ C
(0)
k and let F̃Ψ
(◦
ξ
)
be the folded fan for
◦
ξ then the multiplicity of the
weight ξ is subject to the recursion relation
m
(µ)
ξ
∣∣∣
ξ∈C(0)
k
=
∑
fψ(
◦
ξ)∈F̃Ψ(
◦
ξ)
η̂
(◦
ξ, ξ + fψ
(◦
ξ
))
m
(µ)
ξ+fψ(
◦
ξ)
+ δξ,µ. (6)
4 Folded fans and string functions
For the highest weight module Lµ (g) with µ =
(◦
µ; k; 0
)
of level k consider the set of maximal
vectors belonging to C(0)
k
Zµ
k :=
{
ζ ∈ max(µ) ∩ C(0)
k
}
.
Let π be a projection to the subset of P with level k and grade n = 0 and introduce the set:
Ξµk :=
{
ξ = π ◦ ζ | ζ ∈ Zµ
k
}
.
The cardinality
p(µ)
max := #
(
Ξµk
)
String Functions for Affine Lie Algebras Integrable Modules 7
is finite and we can enumerate the corresponding weights ξj :
Ξµk =
{
ξj | j = 1, . . . , p(µ)
max
}
.
The string functions necessary and sufficient to construct the diagram N µ (and correspondingly
the character ch (Lµ)) are{
σµ,kζ | ζ ∈ Zµ
k
}
, ch (Lµ) =
∑
ξ∈max(µ)
σµξ
(
e−δ
)
eξ =
∑
w∈W/Wζ , ζ∈Zµk
σµ,kζ
(
e−δ
)
ew◦ζ .
Let us consider these string functions as starting from the points ξj rather than from ζ’s.
(For ζ = ξs − lδ ∈ Zµ
k the expansion σµξs (q) =
∞∑
n=0
m
(µ)
(ξs−nδ)q
n starts with string coefficients
m(ξs−nδ)|n<l = 0.) Denote these extended string functions by σµ,kj and introduce the set
Σµ
k :=
{
σµ,kj | ξj ∈ Ξµk
}
.
Let us apply the relation (6) to the weights of the string σµ,kj ∈ Σµ
k and put ξ = ξj + njδ,
m
(µ)
(
◦
ξj ;k;nj)
=
∑
fψ(
◦
ξj)∈F̃Ψ(
◦
ξj)
η̂
( ◦
ξj ,
( ◦
ξj +
◦
fψ
( ◦
ξj
)
; k;nj + n
fψ(
◦
ξj)
))
×m
(µ)(
(
◦
ξj+
◦
fψ(
◦
ξj));k;(nj+n
fψ(
◦
ξj)
)
) + δξj ,µ.
In the folded fan F̃Ψ
( ◦
ξj
)
let us separate the summation over the grades nfψ and the classical
parts
◦
fψ of the shifts fψ
( ◦
ξj
)
. The overcrossing terms vanish because their multiplicities are
zero. The first term in the r.h.s. of the recursion relation takes the form∑
n
fψ(
◦
ξj)
∑
◦
fψ(
◦
ξj);
fψ(
◦
ξj)∈F̃Ψ(
◦
ξj)
η̂
( ◦
ξj ,
( ◦
ξj +
◦
fψ
( ◦
ξj
)
; k, ;nj + n
fψ(
◦
ξj)
))
m
(µ)(
(
◦
ξj+
◦
fψ(
◦
ξj));k;(nj+n
fψ(
◦
ξj)
)
).
For the same reason we can spread the first summation over all the positive grades. It is
sufficient to include the vector with zero coordinates into the folded fan and put the multiplicity
η
(◦
ξ, ξ
)
= −1. Introduce the set
FΨ
( ◦
ξj
)
:= F̃Ψ
( ◦
ξj
)
∪ (0; 0; 0) .
It is called the full folded fan or simply the folded fan when from the context it is clear what fan
F̃Ψ
( ◦
ξj
)
or FΨ
( ◦
ξj
)
is actually used. The set of multiplicities η
(◦
ξ, ξ′
)
for the shifts in FΨ
(◦
ξ
)
is
thus fixed as follows:
η
(◦
ξ, ξ′
)∣∣
ξ′−ξ∈FΨ(
◦
ξ)
:= −
∑
w∈W,
wφ(ξ,w)◦(ξ−(w◦ρ−ρ))=ξ′
ε(w), (7)
and the recursion property (6) is reformulated:∑
fψ(
◦
ξ)∈FΨ(
◦
ξ)
η
(◦
ξ, ξ + fψ
(◦
ξ
))
m
(µ)
ξ+fψ(
◦
ξ)
+ δξ,µ = 0, ξ ∈ C(0)
k .
8 P. Kulish and V. Lyakhovsky
For the string σµ,kj we can rewrite this relation separating the summations:
∞∑
n=0
∑
◦
fψ(
◦
ξj)
fψ(
◦
ξj)∈FΨ(
◦
ξj)
η
( ◦
ξj ,
( ◦
ξj +
◦
fψ
( ◦
ξj
)
; k;nj + n
))
m
(µ)
((
◦
ξj+
◦
fψ(
◦
ξj));k;(nj+n))
+ δξj ,µ = 0.
The properties of N µ for an integrable modules Lµ guarantee that for any finite nj the first sum
is finite. It extends to n ≤ −nj (remember that nj is negative). The second sum can also be
augmented so that the vectors
( ◦
ξj +
◦
fψ
( ◦
ξj
)
; k; 0
)
=
( ◦
ξs; k; 0
)
run over the set Ξµk . Now taking
into account that nj,s does not depend on nj (Lemma 1) the notation can be simplified:
ηj,s (n) := η
( ◦
ξj ,
( ◦
ξs; k;nj + n
))
, m
(µ)
s,nj+n
:= m
(µ)
(
◦
ξs;k;nj+n)
,
and the recursion property for the string functions in
{
σµj |ξj ∈ Ξµk
}
can be stated:
Proposition 2. Let Lµ be the integrable highest weight module of g, µ =
(◦
µ; k; 0
)
, p(µ)
max :=
#
(
Ξµk
)
, ξj =
( ◦
ξj ; k;nj
)
∈ Ξµk + njδ , let FΨ
( ◦
ξj
)
be the full folded fan for
◦
ξj and ηj,s (n) =
−
∑̃
wj,s,
ε(w̃j,s) where the summation is over the elements w̃j,s of W satisfying the equation wφ(ξ,w)◦
(ξj − (w̃j,s ◦ ρ− ρ)) =
( ◦
ξs; k;nj + n
)
, then for the string function coefficients m(µ)
s,nj+n
the fol-
lowing relation holds:
p
(µ)
max∑
s=1
0∑
n=−nj
ηj,s (n)m(µ)
s,nj+n
= −δξj ,µ. (8)
For a fixed nj ≤ 0 consider the sequence of the string weights
ξj;nj =
( ◦
ξj ; k;nj
)
, ξj;nj+1 =
( ◦
ξj ; k;nj + 1
)
, . . . , ξj;0 =
( ◦
ξj ; k; 0
)
,
and write down two (|nj |+ 1)-dimensional vectors: the coordinates of the first one are the
coefficients of the s-th string {σµs },
m(µ)
(s;nj)
:=
(
m(µ)
s,nj ,m
(µ)
s,nj+1, . . . ,m
(µ)
s,0
)
,
the second indicates that the j-th string σµ,kj is starting at the highest weight µ,
δµ(j;nj) := (0, 0, . . . ,−1) .
For the weights with n ≥ nj we have the sequence of relations of the type (8):
p
(µ)
max∑
s=1
−nj∑
n=0
ηj,s (n)m(µ)
s,nj+n
= 0,
p
(µ)
max∑
s=1
−nj−1∑
n=0
ηj,s (n)m(µ)
s,nj+n+1 = 0,
· · · · · · · · · · · · · · · · · · · · · · · · · · · · · · ,
String Functions for Affine Lie Algebras Integrable Modules 9
p
(µ)
max∑
s=1
ηj,s (0)m(µ)
s,0 = −1. (9)
Introduce the upper triangular (|nj |+ 1)× (|nj |+ 1)-matrix
MΞµ
(j,s) :=
ηj,s (0) ηj,s (1) · · · ηj,s (−nj)
0 ηj,s (0) · · · ηj,s (−nj − 1)
...
...
...
...
0 0 · · · ηj,s (0)
.
The set of relations (9) reads:
MΞµ
(j,s) ·m
(µ)
(s;nj)
= δµ(j;nj). (10)
Perform the same procedure for the other weights ξj ∈ Ξµk putting the minimal values of grade
equal: nj |j=1,...,p
(µ)
max
= u, that is construct all the folded fans FΨ
( ◦
ξj
)
(till the grade u) and
the corresponding sets of multiplicities ηj,s (n) (defined by relations (7)). For j = 1, . . . , p(µ)
max
compose (|u|+ 1)2 equations of the type (10):
MΞµ
(j,s)m
(µ)
(s;nj)
= δµ(j;nj), j, s = 1, . . . , p(µ)
max. (11)
Form two (|u|+ 1)× p
(µ)
max-dimensional vectors: the first with the string coefficients,
m(µ)
(u) :=
(
m
(µ)
1,u,m
(µ)
1,u+1, . . . ,m
(µ)
1,0 ,m
(µ)
2,u,m
(µ)
2,u+1, . . . ,m
(µ)
2,0 , . . .
. . . ,m
(µ)
p
(µ)
max,u
,m
(µ)
p
(µ)
max,u+1
, . . . ,m
(µ)
p
(µ)
max,0
)
,
the second indicating that the string σµ,kj with number j starts at the highest weight µ,
δµ(u) := (0, 0, . . . , 0, 0, 0, . . . , 0, 0, 0, . . . ,−1, 0, 0, . . . , 0) ,
(here only in the j-th subsequence the last ((|u|+ 1)-th) coordinate is not zero). Define the
(|u|+ 1) p(µ)
max × (|u|+ 1) p(µ)
max-matrix – the block-matrix with the blocks MΞµ
(j,s):
MΞµ :=
∥∥∥MΞµ
(j,s)
∥∥∥
j,s=1,...,p
(µ)
max
.
In these terms the relations (11) have the following integral form:
MΞµ m(µ)
(u) = δµ(u). (12)
The matrix MΞµ being invertible the equation (12) can be solved. Thus we have demonstrated
that the strings σµ,kj are determined by the matrix MΞµ whose elements are the full folded fan
weight multiplicities:
Proposition 3. Let Lµ be an integrable highest weight module of g, µ =
(◦
µ; k; 0
)
, p(µ)
max :=
#
(
Ξµk
)
, ξj =
( ◦
ξj ; k;nj
)
∈ Ξµk + njδ; let FΨ
( ◦
ξj
)
be the full folded fan for
◦
ξj and MΞµ – the
(|nj |+ 1) p(µ)
max × (|nj |+ 1) p(µ)
max-matrix formed by the blocks MΞµ
(j,s)
MΞµ
(j,s) :=
ηj,s (0) ηj,s (1) · · · ηj,s (−nj)
0 ηj,s (0) · · · ηj,s (−nj − 1)
...
...
...
...
0 0 · · · ηj,s (0)
10 P. Kulish and V. Lyakhovsky
where the elements ηj,s (n) are the multiplicities of the folded fan weights,
ηj,s (n) = −
∑
w̃j,s,
ε(w̃j,s)
with the summation over the elements w̃j,s ∈W satisfying the equation
wφ(ξ,w) ◦ (ξj − (w̃j,s ◦ ρ− ρ)) =
( ◦
ξs; k;nj + n
)
.
Let the string function coefficients be the coordinates in the nj + 1-subsequences of the vec-
tor m(µ)
(nj)
. Then for the coefficients of
{
σµ,kj | j = 1, . . . , p(µ)
max
}
the following relation folds:
m(µ)
(nj)
=
(
MΞµ
)−1
δµ(nj). (13)
Thus the solution m(µ)
(nj)
describes all the string functions relevant to the chosen module Lµ
(with the grades no less than the preliminary fixed nj = u).
To describe the complete string functions it is sufficient to send u to the limit u→ −∞ .
5 Examples
5.1 g = A
(1)
2
Consider the fan Γh⊂g (with nψ(0) ≤ 9):
Γh⊂g =
{
(0, 1, 0, 0, 1) , (2, 1, 0, 0,−1) , (1, 0, 0, 0, 1) , (1, 2, 0, 0,−1) ,
(2, 2, 0, 0, 1) , (3, 1, 0, 1, 1) , (−1, 1, 0, 1,−1) , (1, 3, 0, 1, 1) ,
(1,−1, 0, 1,−1) , (3, 3, 0, 1,−1) , (−1,−1, 0, 1, 1) , (3, 4, 0, 2, 1) ,
(0,−2, 0, 2, 1) , (2, 4, 0, 2,−1) , (−1,−2, 0, 2,−1) , (4, 3, 0, 2, 1) ,
(−2, 0, 0, 2, 1) , (4, 2, 0, 2,−1) , (−2,−1, 0, 2,−1) , (0, 3, 0, 2,−1) ,
(3, 0, 0, 2,−1) , (−1, 2, 0, 2, 1) , (2,−1, 0, 2, 1) , (0, 4, 0, 4, 1) ,
(−3,−2, 0, 4, 1) , (5, 4, 0, 4,−1) , (2,−2, 0, 4,−1) , (4, 0, 0, 4, 1) ,
(−2,−3, 0, 4, 1) , (4, 5, 0, 4,−1) , (−2, 2, 0, 4,−1) , (−3, 0, 0, 4,−1) ,
(1,−3, 0, 5, 1) , (5, 1, 0, 5,−1) , (5, 5, 0, 5, 1) , (1, 5, 0, 5,−1) ,
(0,−3, 0, 4,−1) , (2, 5, 0, 4, 1) , (5, 2, 0, 4, 1) , (−3,−3, 0, 5,−1) ,
(−3, 1, 0, 5, 1) , (6, 4, 0, 6, 1) , (3,−2, 0, 6, 1) , (−1, 4, 0, 6,−1) ,
(−4,−2, 0, 6,−1) , (4, 6, 0, 6, 1) , (−2, 3, 0, 6, 1) , (4,−1, 0, 6,−1) ,
(−2,−4, 0, 6,−1) , (3, 6, 0, 6,−1) , (6, 3, 0, 6,−1) , (−4,−1, 0, 6, 1) ,
(−1,−4, 0, 6, 1) , (6, 1, 0, 8, 1) , (−4, 1, 0, 8,−1) , (1, 6, 0, 8, 1) ,
(1,−4, 0, 8,−1) , (6, 6, 0, 8,−1) , (−4,−4, 0, 8, 1) , (3, 7, 0, 9, 1) ,
(−3,−5, 0, 9, 1) , (5, 7, 0, 9,−1) , (−1,−5, 0, 9,−1) , (7, 3, 0, 9, 1) ,
(−5,−3, 0, 9, 1) , (7, 5, 0, 9,−1) , (−5,−1, 0, 9,−1) , (−3, 3, 0, 9,−1) ,
(3,−3, 0, 9,−1) , (−1, 5, 0, 9, 1) , (5,−1, 0, 9, 1) , . . .
}
. (14)
Here the first two coordinates are classical in the basis of simple roots {α1, α2}, next comes
the level k = 0, the grade nψ(0) and the multiplicity mψ(0) of the weight ψ(0) ∈ Γh⊂g (for the
injection h −→ g we have mψ(0) = −ε(w)).
String Functions for Affine Lie Algebras Integrable Modules 11
5.1.1 k = 1
The set C(0)
1;0 contains three weights (p(µ)
max = 3):
C
(0)
1;0 =
{
(0, 0; 1; 0) , (
◦
ω1; 1; 0), (
◦
ω2; 1; 0)
}
= {ω0, ω1, ω2}
=
{
(0, 0; 1; 0) , (2/3, 1/3; 1; 0) , (1/3, 2/3; 1; 0)
}
,
ωi are the fundamental weights.
The classical components
◦
fψ of the folded fan shifts
wφ(ξ,w) ◦ (ξ − (w ◦ ρ− ρ))− ξ, ξ ∈ C(0)
k
belong to the classical root lattice Q
(◦
g
)
. For any weight ξ =
(◦
ξ; 1; 0
)
∈ C
(0)
1;0 these classical
components are equal to zero, thus the folded fan has the form
FΨ
( ◦
ξj
)
:=
{(
0; 0;n
fψ(
◦
ξ)
)}
, ξj ∈ C(0)
k , j = 1, 2, 3.
It is convenient to indicate the multiplicities
ηj,s (n) = −
∑
w̃j,s∈W,
wφ(ξ,w)◦(ξj−(w̃j,s◦ρ−ρ))=(
◦
ξs;k;nj+n)
ε(w̃j,s)
as the additional coordinates of the shifts fψ:
FΨ
( ◦
ξj
)
:=
{(
0, 0, n
fψ(
◦
ξ)
, ηj,s
(
n
fψ(
◦
ξ)
))}
.
Thus any folded fan for the highest weight µ of level k = 1 contains only “one string”. Moreover
the fans FΨ
( ◦
ξj
)
do not depend on the choice of ξj =
( ◦
ξj ; 1; 0
)
∈ C(0)
1;0 . The latter results are in
full accord with the Proposition 12.6 in [6].
Using the fan Γh⊂g we obtain the folded fan (only the shifts with nonzero multiplicities ηj,j
are indicated, the maximal grade here is n = 20):
FΨ
( ◦
ξj
)
:=
{
(0; 0; 0;−1) , (0; 0; 1; 2) , (0; 0; 2; 1) , (0; 0; 3;−2) , (0; 0; 4;−1) ,
(0; 0; 5;−2) , (0; 0; 7; 2) , (0; 0; 8; 2) , (0; 0; 9;−1) , (0; 0; 10;−1) , (0; 0; 13;−2) ,
(0; 0; 14;−3) , (0; 0; 15; 2) , (0; 0; 16;−2) , (0; 0; 19; 2) , (0; 0; 20; 2) , . . .
}
The multiplicities
{ηj,j (n)}n=0,...,20 =
{
− 1, 2, 1,−2,−1,−2, 2, 0, 2, 2,−1,−1, 0, 0,−2,−3, 2,−2, 0, 0, 2, 2
}
form the unique nonzero matrix M(j,j) for j = 1, 2, 3:
M(j,j) :=
ηj,j (0) ηj,j (1) · · · ηj,j (−nj)
0 ηj,j (0) · · · ηj,j (−nj − 1)
...
...
...
...
0 0 · · · ηj,j (0)
.
The matrix M is block-diagonal and the equation (13) splits into three equivalent (for µ =
(0, 0; 1; 0) , (2/3, 1/3; 1; 0) , (1/3, 2/3; 1; 0)) relations m(µ)
(j;−20) = M−1
(j,j)δ
µ
(j;−20) determining the
unique string function with coefficients m(µ)
(j;−20) =
(
m
(µ)
j,−20,m
(µ)
j,−19, . . . ,m
(µ)
j,0
)
,
σ (q) = 1 + 2q + 5q2 + 10q3 + 20q4 + 36q5 + 65q6 + 110q7 + 185q8 + 300q9 + 481q10
12 P. Kulish and V. Lyakhovsky
+ 752q11 + 1165q12 + 1770q13 + 2665q14 + 3956q15 + 5822q16 + 8470q17
+ 12230q18 + 17490q19 + 24842q20 + · · · .
The obtained expression coincides with the expansion of the square of the inverse Euler function
(see Proposition 12.13 in [6] and the relation (12.13.4) there).
5.1.2 k = 2
The set C(0)
1;0 contains six weights:
C
(0)
1;0 =
{
(0, 0; 2; 0) , (
◦
ω1; 2; 0), (
◦
ω2; 2; 0),
(
◦
ω1 +
◦
ω2; 2; 0), (2
◦
ω1; 2; 0), (2
◦
ω2; 2; 0)
}
=
=
{
(0, 0; 2; 0) , (2/3, 1/3; 2; 0) , (1/3, 2/3; 2; 0) ,
(1, 1; 2; 0) , (4/3, 2/3; 2; 0) , (2/3, 4/3; 2; 0)
}
.
This set is divided into 3 congruence classes. The fan shifts cannot connect vectors from different
classes. Thus instead of the set Ξ2 we can consider three subsets separately:
Ξ2;I = {(0, 0; 2; 0) , (1, 1; 2; 0)} ,
Ξ2;II = {(2/3, 1/3; 2; 0) , (2/3, 4/3; 2; 0)} ,
Ξ2;III = {(1/3, 2/3; 2; 0) , (4/3, 2/3; 2; 0)} .
Let us start with
◦
ξs ∈ Ξ2;I and µ = (0, 0; 2; 0). Here we have two folded fans FΨ
( ◦
ξ1
)
and
FΨ
( ◦
ξ2
)
.
Using the fan Γh⊂g (14) we obtain the folded fans (the maximal grade here is n = 9):
FΨ
( ◦
ξ1
)
:=
{
(0; 0; 0;−1) , (0; 0; 2; 1) , (0; 0; 4; 2) , (0; 0; 8;−2) , (0; 0; 10;−2) ,
(1; 1; 0; 2) , (1; 1; 1;−1) , (1; 1; 2;−2) , (1; 1; 3;−2) , (1; 1; 4; 2) ,
(1; 1; 5; 1) , (1; 1; 6;−2) , (1; 1; 7; 2) , (1; 1; 9;−1) , . . .
}
,
FΨ
( ◦
ξ2
)
:=
{
(0; 0; 1; 1) , (0; 0; 3;−2) , (0; 0; 7; 1) , (0; 0; 9;−2) ,
(1; 1; 1; 2) , (1; 1; 2;−2) , (1; 1; 4; 1) , (1; 1; 5; 2) , (1; 1; 6; 2) ,
(1; 1; 7;−2) , (1; 1; 8;−2) , (1; 1; 9;−2) , . . .
}
.
The multiplicities (n = 0, . . . , 10)
{η1,1 (−10 + n)} = {−1, 0, 1, 0, 2, 0, 0, 0,−2, 0,−2} ,
{η1,2 (−10 + n)} = {2,−1,−2,−2, 2, 1,−2, 2, 0,−1, 0} ,
{η2,1 (−10 + n)} = {0, 1, 0,−2, 0, 0, 0, 1, 0,−1, 0} ,
{η2,2 (−10 + n)} = {−1, 2,−2, 0, 1, 2, 2,−2,−2,−2, 0} ,
form the matrices M(Ξ,2;v)
(s,t) for s, t = 1, 2:
M(Ξ,2;v)
(s,t) :=
ηs,t (0) ηs,t (1) · · · ηs,t (10)
0 ηs,t (0) · · · ηs,t (9)
...
...
...
...
0 0 · · · ηs,t (0)
.
String Functions for Affine Lie Algebras Integrable Modules 13
The block-matrix M(Ξ,2;v) is
M(Ξ,2;v) :=
∥∥∥∥∥ M(Ξ,2;v)
(1,1) M(Ξ,2;v)
(1,2)
M(Ξ,2;v)
(2,1) M(Ξ,2;v)
(2,2)
∥∥∥∥∥ .
The equation
m(0,0;2;0)
(−10) =
(
M(Ξ,2;v)
)−1
δ
(0,0;2;0)
(−10)
gives two string functions σ(0,0;2;0)
(s;−10) with the coefficients in the subsections of the vector m(0,0;2;0)
(−10) :
σ
(0,0;2;0)
(1;−10) = 1 + 2q + 8q2 + 20q3 + 52q4 + 116q5
+ 256q6 + 522q7 + 1045q8 + 1996q9 + 3736q10 + · · · ,
σ
(0,0;2;0)
(2;−10) = q + 4q2 + 12q3 + 32q4 + 77q5
+ 172q6 + 365q7 + 740q8 + 1445q9 + 2736q10 + · · · .
In the second congruence class Ξ2;II= {(2/3, 1/3; 2; 0) , (2/3, 4/3; 2; 0)} put µ=(2/3, 1/3; 2; 0).
Again we have two folded fans FΨ
( ◦
ξ1
)
and FΨ
( ◦
ξ2
)
.
The multiplicities (n = 0, . . . , 10):
{η1,1 (−10 + n)} = {−1, 2,−2, 0, 1, 2, 2,−2,−2,−2, 0} ,
{η1,2 (−10 + n)} = {1, 0,−2, 0, 0, 0, 1, 0,−1, 0, 2} ,
{η2,1 (−10 + n)} = {0, 2,−1,−2,−2, 2, 1,−2, 2, 0,−1} ,
{η2,2 (−10 + n)} = {−1, 0, 1, 0, 2, 0, 0, 0,−2, 0,−2} .
form the matrices MΞ2;II
(s,t) for s, t = 1, 2 and the 22× 22 block-matrix M
MΞ2;II :=
∥∥∥∥∥ MΞ2;II
(1,1) MΞ2;II
(1,2)
MΞ2;II
(2,1) MΞ2;II
(2,2)
∥∥∥∥∥ .
The equation
m(2/3,1/3;2;0)
(−10) =
(
MΞ2;II
)−1
δ
(2/3,1/3;2;0)
(−10)
gives two string functions σ(2/3,1/3;2;0)
(s;−10) for the module L(2/3,1/3;2;0) with the coefficients in the
subsections of the vector m(2/3,1/3;2;0)
(−10) :
σ
(2/3,1/3;2;0)
(1;−10) = 1 + 4q + 13q2 + 36q3 + 89q4 + 204q5
+ 441q6 + 908q7 + 1798q8 + 3444q9 + 6410q10 + · · · ,
σ
(2/3,1/3;2;0)
(2;−10) = 2q + 7q2 + 22q3 + 56q4 + 136q5
+ 300q6 + 636q7 + 1280q8 + 2498q9 + 4708q10 + · · · .
For the third congruence class Ξ2;III= {(1/3, 2/3; 2; 0) , (4/3, 2/3; 2; 0)} the folded fans FΨ
( ◦
ξ1
)
and FΨ
( ◦
ξ2
)
are the same as for the second one. As a result the string functions also coincide:
σ
(1/3,2/3;2;0)
(s;−10) = σ
(2/3,1/3;2;0)
(s;−10) in accord with the A2 external automorphism.
14 P. Kulish and V. Lyakhovsky
5.1.3 k = 4
The set C(0)
1;0 contains 15 projected maximal weights
{ξj | ξj ∈ Ξ4; j = 1, . . . , pmax = 15} ,
C
(0)
1;0 =
4ω0, 3ω0 + ω1, 3ω0 + ω2, 2ω0 + 2ω1, 2ω0 + 2ω2,
2ω0 + ω1 + ω2, ω0 + 3ω1, ω0 + 3ω2, ω0 + 2ω1 + ω2,
ω0 + ω1 + 2ω2, 3ω1 + ω2, ω1 + 3ω2, 2ω1 + 2ω2, 4ω1, 4ω2
.
This set is divided into 3 congruence classes. Instead of the set Ξ4 we can consider separately
three subsets:
Ξ4;I = {(0, 0; 4; 0) , (1, 1; 4; 0) , (1, 2; 4; 0) , (2, 1; 4; 0) , (2, 2; 4; 0)} ,
Ξ4;II =
{
(2/3, 1/3; 4; 0) , (2/3, 4/3; 4; 0) , (5/3, 4/3; 4; 0) , (5/3, 7/3; 4; 0) , (8/3, 4/3; 4; 0)
}
,
Ξ4;III =
{
(1/3, 2/3; 4; 0) , (4/3, 2/3; 4; 0) , (4/3, 5/3; 4; 0) , (7/3, 5/3; 4; 0) , (4/3, 8/3; 4; 0)
}
.
Let us start with
◦
ξs ∈ Ξ4;I and µ = (0, 0; 4; 0). Here we have 5 folded fans FΨ
( ◦
ξs
)
, s = 1, . . . , 5.
Using the fan Γh⊂g (14) we construct the folded fans (the maximal grade here is chosen to
be n = 9):
FΨ
( ◦
ξ1
)
:=
{
(0, 0; 0;−1) , (0, 0; 9; 2) , (1, 1; 0; 2), (1, 1; 1; 1), (1, 1; 3;−1),
(1, 1; 4;−2), (1, 1; 5; 2), (1, 1; 6;−2), (1, 1; 7;−1), (1, 1; 8; 2),
(1, 2; 0;−1), (1, 2; 1;−1), (1, 2; 3; 1), (1, 2; 5; 1), (1, 2; 8; 1),
(2, 1; 0;−1), (2, 1; 1;−1), (2, 1; 3; 1), (2, 1; 5; 1), (2, 1; 8; 1),
(2, 2; 0; 1), (2, 2; 2; 2), (2, 2; 4;−2), (2, 2; 6;−2), (2, 2; 8;−2), . . .
}
,
FΨ
( ◦
ξ2
)
:=
{
(0, 0; 1; 1) , (0; 0; 5;−1) , (1; 1; 0;−1) , (1; 1; 2;−1) , (1; 1; 4; 2) , (1; 1; 5;−2) ,
(1; 1; 8; 2) , (1; 1; 9; 2) , (1; 1; 1; 2) , (1; 1; 2;−2) , (1; 1; 4; 1) , (1; 1; 5; 2) ,
(1; 1; 6; 2) , (1; 2; 0; 1) , (1; 2; 1;−1) , (1; 2; 2; 1) , (1; 2; 4; 1) , (1; 2; 5;−1) ,
(1; 2; 6;−1) , (1; 2; 7;−1) , (2; 1; 0; 1) , (2; 1; 1;−1) , (2; 1; 2; 1) , (2; 1; 4; 1) ,
(2; 1; 5;−1) , (2; 1; 6;−1) , (2; 1; 7;−1) , (2; 2; 0; 1) , (2; 2; 2; 2) , (2; 2; 4;−2) ,
(2; 2; 6;−2) , (2; 2; 8;−2) , . . .
}
,
FΨ
( ◦
ξ3
)
:=
{
(0, 0; 2;−1) , (0; 0; 6; 1) , (1; 1; 1; 1) , (1; 1; 4; 2) , (1; 1; 6;−2) , (1; 1; 7;−2) ,
(1; 2; 0;−1) , (1; 2; 1; 1) , (1; 2; 4;−1) , (1; 1; 5;−1) , (1; 2; 6; 1) , (1; 2; 8; 1) ,
(1; 2; 9; 2) , (2; 1; 1;−1) , (2; 1; 2;−1) , (2; 1; 3; 2) , (2; 1; 5; 1) , (2; 1; 6;−1) ,
(2; 1; 8;−1) , (2; 2; 0; 1) , (2; 2; 2;−1) , (2; 2; 8; 1) , . . .
}
.
The fan FΨ
( ◦
ξ4
)
is equal to
{
FΨ
( ◦
ξ3
)
| (1; 2;n;m)
(2; 1;n;m)
}
FΨ
( ◦
ξ5
)
:=
{
(0, 0; 4; 1) , (0; 0; 8;−2) , (1; 1; 1; 1) , (1; 1; 2;−2) ,
(1; 1; 3;−2) , (1; 1; 4; 2) , (1; 1; 5; 1) , (1; 1; 7;−1) , (1; 1; 8; 2) ,
(1; 2; 1; 1) , (1; 2; 2;−1) , (1; 2; 6;−1) , (1; 2; 7; 1) , (1; 2; 9; 1) ,
(2; 1; 1; 1) , (2; 1; 2;−1) , (2; 1; 6;−1) , (2; 1; 7; 1) , (2; 1; 9; 1) ,
(2; 2; 0;−1) , (2; 2; 2; 2) , (2; 2; 6;−2) , (2; 2; 8;−1) , . . .
}
.
Their multiplicities (for n = 0, . . . , 9)
{η1,1 (−9 + n)} = {−1, 0, 0, 0, 0, 0, 0, 0, 2, 0} ,
String Functions for Affine Lie Algebras Integrable Modules 15
{η1,2 (−9 + n)} = {2, 1, 0,−1,−2, 2,−2,−1, 2, 0} ,
{η1,3 (−9 + n)} = {−1,−1, 0, 1, 0, 1, 0, 0, 1, 0} ,
{η1,4 (−9 + n)} = {−1,−1, 0, 1, 0, 1, 0, 0, 1, 0} ,
{η1,5 (−9 + n)} = {1, 0, 2, 0,−2, 0,−2, 0,−2, 0} ,
{η2,1 (−9 + n)} = {0, 1, 0, 0, 0,−1, 0, 0, 0, 0} ,
{η2,2 (−9 + n)} = {−1, 0,−1, 0, 2,−2, 0, 0, 2, 2} ,
{η2,3 (−9 + n)} = {1,−1, 1, 0, 1,−1,−1,−1, 0, 0} ,
{η2,4 (−9 + n)} = {1,−1, 1, 0, 1,−1,−1,−1, 0, 0} ,
{η2,5 (−9 + n)} = {1, 0, 2, 0,−2, 0,−2, 0,−2, 0} ,
{η3,1 (−9 + n)} = {0, 0,−1, 0, 0, 0, 1, 0, 0, 0} ,
{η3,2 (−9 + n)} = {0, 1, 0, 0, 2, 0,−2,−2, 0, 0} ,
{η3,3 (−9 + n)} = {−1, 1, 0, 0,−1,−1, 1, 0, 1, 2} ,
{η3,4 (−9 + n)} = {0,−1,−1, 2, 0, 1,−1, 0,−1, 0} ,
{η3,5 (−9 + n)} = {1, 0,−1, 0, 0, 0, 0, 0, 1, 0} ,
{η4,1 (−9 + n)} = {0, 0,−1, 0, 0, 0, 1, 0, 0, 0} ,
{η4,2 (−9 + n)} = {0, 1, 0, 0, 2, 0,−2,−2, 0, 0} ,
{η4,3 (−9 + n)} = {0,−1,−1, 2, 0, 1,−1, 0,−1, 0} ,
{η4,4 (−9 + n)} = {−1, 1, 0, 0,−1,−1, 1, 0, 1, 2} ,
{η4,5 (−9 + n)} = {1, 0,−1, 0, 0, 0, 0, 0, 1, 0} ,
{η5,1 (−9 + n)} = {0, 0, 0, 0, 1, 0, 0, 0,−2, 0} ,
{η5,2 (−9 + n)} = {0, 1,−2,−2, 2, 1, 0,−1, 2, 0} ,
{η5,3 (−9 + n)} = {0, 1,−1, 0, 0, 0,−1, 1, 0, 1} ,
{η5,4 (−9 + n)} = {0, 1,−1, 0, 0, 0,−1, 1, 0, 1} ,
{η5,5 (−9 + n)} = {−1, 0, 2, 0, 0, 0,−2, 0,−1, 0} .
The matrices MΞ4;I
(s,t) for s, t = 1, . . . , 5:
MΞ4;I
(s,t) :=
ηs,t (0) ηs,t (1) · · · ηs,t (9)
0 ηs,t (0) · · · ηs,t (8)
...
...
...
...
0 0 · · · ηs,t (0)
.
For example,
MΞ4;I
(2,3) :=
1 −1 1 0 1 −1 −1 −1 0 0
0 1 −1 1 0 1 −1 −1 −1 0
0 0 1 −1 1 0 1 −1 −1 −1
0 0 0 1 −1 1 0 1 −1 −1
0 0 0 0 1 −1 1 0 1 −1
0 0 0 0 0 1 −1 1 0 1
0 0 0 0 0 0 1 −1 1 0
0 0 0 0 0 0 0 1 −1 1
0 0 0 0 0 0 0 0 1 −1
0 0 0 0 0 0 0 0 0 1
.
Matrices MΞ4;I
(s,t) form the block-matrix MΞ4;I=
∥∥∥MΞ4;I
(s,t)
∥∥∥
s,t=1,...,5
. With this matrix we can de-
scribe five modules of the level 4 with the highest weights µs ∈ Ξ4;I = {(0, 0; 4; 0) , (1, 1; 4; 0) ,
16 P. Kulish and V. Lyakhovsky
(1, 2; 4; 0) , (2, 1; 4; 0) , (2, 2; 4; 0)} . We construct five sets of string functions σ(µs)
(t;−9) in terms of
their coefficients obtained as ten dimensional subsections of the vector m(µs)
(−9):
m(µs)
(−9) =
(
MΞ4;I
)−1
δ
(µs)
(−9).
The answer is as follows:
σ
(0,0;4;0)
(1;−9) = 1 + 2q + 8q2 + 24q3 + 72q4 + 190q5 + 490q6
+ 1176q7 + 2729q8 + 6048q9 + · · · ,
σ
(0,0;4;0)
(2;−9) = q + 4q2 + 15q3 + 48q4 + 138q5 + 366q6 + 913q7 + 2156q8 + 4874q9 + · · · ,
σ
(0,0;4;0)
(3;−9) = q2 + 6q3 + 23q4 + 74q5 + 2121q6 + 556q7 + 1366q8 + 3184q9 + · · · ,
σ
(0,0;4;0)
(4;−9) = q2 + 6q3 + 23q4 + 74q5 + 2121q6 + 556q7 + 1366q8 + 3184q9 + · · · ,
σ
(0,0;4;0)
(5;−9) = q2 + 4q3 + 18q4 + 56q5 + 167q6 + 440q7 + 1103q8 + 2588q9 + · · · ,
σ
(1,1;4;0)
(1;−9) = 2 + 10q + 40q2 + 133q3 + 398q4 + 1084q5 + 2760q6
+ 6632q7 + 15214q8 + 33508q9 + · · · ,
σ
(1,1;4;0)
(2;−9) = 1 + 6q + 27q2 + 96q3 + 298q4 + 836q5 + 2173q6
+ 5310q7 + 12341q8 + 27486q9 + · · · ,
σ
(1,1;4;0)
(3;−9) = 2q2 + 12q3 + 49q4 + 166q5 + 494q6 + 1340q7 + 3387q8 + 8086q9 + · · · ,
σ
(1,1;4;0)
(4;−9) = 2q2 + 12q3 + 49q4 + 166q5 + 494q6 + 1340q7 + 3387q8 + 8086q9 + · · · ,
σ
(1,1;4;0)
(5;−9) = q + 8q2 + 35q3 + 124q4 + 379q5 + 1052q6 + 2700q7 + 6536q8 + 15047q9 + · · · ,
σ
(1,2;4;0)
(1;−9) = 1 + 8q + 32q2 + 110q3 + 322q4 + 872q5 + 2183q6
+ 5186q7 + 11730q8 + 25552q9 + · · · ,
σ
(1,2;4;0)
(2;−9) = 1 + 6q + 25q2 + 85q3 + 255q4 + 695q5 + 1764q6
+ 4226q7 + 9653q8 + 21179q9 + · · · ,
σ
(1,2;4;0)
(3;−9) = 1 + 4q + 16q2 + 54q3 + 163q4 + 450q5 + 1161q6 + 2824q7
+ 6549q8 + 14572q9 + · · · ,
σ
(1,2;4;0)
(4;−9) = 2q + 11q2 + 44q3 + 143q4 + 414q5 + 1096q6 + 2714q7
+ 6364q8 + 14272q9 + · · · ,
σ
(1,2;4;0)
(5;−9) = 2q + 9q2 + 36q3 + 115q4 + 336q5 + 890q6 + 2224q7 + 5241q8 + 11840q9 + · · · .
The next set of string functions σ(2,1;4;0)
(s;−9) coincides with the previous one where the third and the
fourth strings are interchanged: σ(2,1;4;0)
(3;−9) = σ
(1,2;4;0)
(4;−9) , σ
(2,1;4;0)
(4;−9) = σ
(1,2;4;0)
(3;−9) . The last set describes
the module Lµ5 where µ5 is the highest weight in Ξ4;I:
σ
(2,2;4;0)
(1;−9) = 3 + 14q + 58q2 + 184q3 + 536q4 + 1408q5 + 3492q6
+ 8160q7 + 18299q8 + 39428q9 + · · · ,
σ
(2,2;4;0)
(2;−9) = 2 + 11q + 44q2 + 145q3 + 424q4 + 1133q5 + 2830q6
+ 6688q7 + 15102q8 + 32805q9 + · · · ,
String Functions for Affine Lie Algebras Integrable Modules 17
σ
(2,2;4;0)
(3;−9) = 1 + 6q + 25q2 + 86q3 + 260q4 + 716q5 + 1833q6 + 4426q7
+ 10183q8 + 22488q9 + · · · ,
σ
(2,2;4;0)
(4;−9) = 1 + 6q + 25q2 + 86q3 + 260q4 + 716q5 + 1833q6 + 4426q7
+ 10183q8 + 22488q9 + · · · ,
σ
(2,2;4;0)
(5;−9) = 1 + 4q + 19q2 + 64q3 + 202q4 + 560q5 + 1464q6 + 3568q7
+ 8315q8 + 18512q9 + · · · .
Notice that in the congruence class Ξ4;I we have only 17 different string functions.
6 Conclusions
The folded fans FΨ
( ◦
ξj
)
(for a fixed level k and the congruence class Ξk;v of weights in C
(0)
k )
were constructed by transporting to the fundamental Weyl chamber the standard set Ψ̂(0) – the
set of singular weights of module L0 supplied with the anomalous multiplicities. We have found
out that the shifts fψ
(◦
ξ
)
∈ FΨ
(◦
ξ
)
(connecting ξj ∈ Ξk;v) together with their multiplicities ηj,s
describe the recursive properties of the weights of modules Lξj with the highest weights ξj .
Thus the set
{
FΨ
( ◦
ξj
)
| ξj ∈ Ξk;v
}
describes the recursive properties of the string functions{
σµ,kj |µ, ξj ∈ Ξk;v
}
. When for a fixed module Lµ these properties are simultaneously considered
for
{
σµ,kj |µ, ξj ∈ Ξk;v
}
they can be written in a form of the equation MΞ,k;vm(µ)
(u) = δµ(u). In this
equation MΞ,k;v is a matrix formed by the multiplicities ηj,s of the fan shifts, δµ(u) indicates what
weight in the set Ξk;v is chosen to be the highest weight µ of the module and m(µ)
(u) is a vector of
string functions coefficients. As far as MΞ,k;v is invertible the solution m(µ)
(u) =
(
MΞ,k;v
)−1
δµ(u)
can be explicitly written and the full set of string functions
{
σµ,kj |µ, ξj ∈ Ξk;v
}
for Lµ is
determined by this linear equation (at least for any common finite “length” of all the strings).
There are two points that we want to stress. The first is that in this algorithm the singular
vectors ψ ∈ Ψ̂(µ) of Lµ are not needed (except the highest weight µ). The second point is
that the crossections FΨ
( ◦
ξj
)
∩ C(0)
k,0 form the parts of the classical folded fans for
◦
g. It can be
easily verified that the string starting vectors
{
σµ,kj |µ, ξj ∈ Ξk;v; n = 0
}
and their multiplicities
present the diagram N
◦
µ∩C(0)
k of the module L
◦
µ
(◦
g
)
. In general the crossections FΨ
( ◦
ξj
)
∩C(0)
(◦
g
)
do not coincide with the classical folded fans because the chambers C(0)
(◦
g
)
are infinite (contrary
to C(0)
k,0 for any finite k).
As it was demonstrated in the examples the folded fans provide an effective tool when study-
ing the string functions for integrable highest weight modules of affine Lie algebras.
Acknowledgements
The authors appreciate helpful remarks made by the Referees. The work was supported in part
by RFBR grants N 06-01-00451, N 08-01-00638 and the National Project RNP.2.1.1.1112.
18 P. Kulish and V. Lyakhovsky
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http://arxiv.org/abs/hep-th/9605187
http://arxiv.org/abs/hep-th/0410105
http://arxiv.org/abs/hep-th/0407277
http://arxiv.org/abs/math-ph/0505037
http://arxiv.org/abs/hep-th/9408087
http://arxiv.org/abs/0707.1635
http://arxiv.org/abs/0812.2124
1 Introduction
2 Basic definitions and relations
3 Folding a fan
4 Folded fans and string functions
5 Examples
5.1 g=A_2^{(1)}
5.1.1 k=1
5.1.2 k=2
5.1.3 k=4
6 Conclusions
References
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