KKR type bijection for the exceptional affine algebra E_6^{(1)}

Abstract

For the exceptional affine type E_6^{(1)} we establish a statistic-preserving
bijection between the highest weight paths consisting of the simplest
Kirillov-Reshetikhin crystal and the rigged configurations. The algorithm only
uses the structure of the crystal graph, hence could also be applied to other
exceptional types.

Full text

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arXiv:1105.1636v2 [math.QA] 10 Jun 2011
KKR TYPE BIJECTION FOR THE EXCEPTIONAL AFFINE
ALGEBRA E(1)
6
MASATO OKADO AND NOBUMASA SANO
Abstract. For the exceptional affine type E(1)
bijection between the highest weight paths consisting of the simplest Kirillov-
Reshetikhin crystal and the rigged configurations. The algorithm only uses the
structure of the crystal graph, hence could also be applied to other exceptional
types.
6
we establish a statistic-preserving
1. Introduction
In a pioneering work [15] Kerov, Kirillov and Reshetikhin introduced a new com-
binatorial object, called rigged configuration, through Bethe ansatz analysis of the
Heisenberg spin chain, and constructed a bijection between rigged configurations
and semistandard tableaux. One of the amazing properties of the rigged config-
uration is that it possesses a natural statistic and the statistic coincides with the
charge by Lascoux and Sch¨ utzenberger [21] on the tableau under the bijection.
Subsequently, Nakayashiki and Yamada [25] studied the meaning of the charge in
terms of Kashiwara’s crystal bases. They considered the crystal base Blof the l-fold
symmetric tensor representation of the n-dimensional irreducible Uq(?sln)-module.
tion, is defined via the q → 0 limit of the quantum R-matrix. Using this H they
constructed a function D on the multiple tensor product Bl1⊗ ··· ⊗ Blm. They
then showed that under a certain bijection sending highest weight vectors or paths
of Bl1⊗···⊗Blmto semistandard tableaux, the value of D agrees with the charge,
thereby proving that the well-known Kostka polynomial is represented as a gener-
ating function of highest weight paths with statistic D. This generating function
is denoted by X and the one of rigged configurations by M. The equality X = M
was extended to the most general case for affine type A in [17]. See also [30] for
review.
It did not take long before this kind of equality was conjectured to exist for
other affine types. For the X side, crystal bases for some finite-dimensional mod-
ules, which are now called Kirillov-Reshetikhin (KR) modules, for quantum affine
algebras have been discovered in [13]. For the M side, the existence of KR mod-
ules were conjectured and a formula to count the number of rigged configurations
were presented in [16]. Introducing an appropriate q-analogue for the formula, the
X = M conjecture [8, 7] was presented. Imitating the one by KKR a bijection
between rigged configurations and highest weight paths consisting of elements of
KR crystals for other nonexceptional affine types was subsequently constructed
in [28, 29, 31]. We note that these bijections have an important application for
For the tensor product Bl⊗ Bl′ an integer-valued function H, called energy func-
2010 Mathematics Subject Classification.
81R50 81R10 05E10 11B65.
Date: May 20, 2011.
Primary 17B37 82B23 05A19; Secondary 17B25
1

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◦0
◦6
◦1
◦2
◦3
◦45
◦
Figure 1. Dynkin diagram for E(1)
6
the analysis of the ultra-discrete integrable systems, also called box-ball systems
[4, 6, 9]. In such systems rigged configurations give the complete set of the action
and angle variables [19, 20].
In this paper we consider the exceptional affine algebra of type E(1)
crystal we deal with is the simplest one denoted in our notation by B1,1, whose
crystal structure was revealed in [22, 5]. We construct a map Φ from rigged con-
figurations to highest weight elements of (B1,1)⊗Lby executing a fundamental
procedure δ repeatedly. We then show Φ is a statistic-preserving bijection (The-
orem 3.2). It is worth mentioning that our procedure only uses the crystal graph
structure of the KR crystal B1,1, hence similar constructions could be possible for
other exceptional types.
We remark that recently Naoi [26] solved, with the help of the results in [3]
and [23], the X = M conjecture for all untwisted affine types when the tensor
product of KR crystals is of the form Br1,1⊗···⊗Brl,1by showing both X and M
are equal to the graded character of a Weyl module, a finite-dimensional current
algebra representation defined in [2]. Hence his result includes ours as a special
case. However, we think our direct method is also important, since it could also be
used for more general cases by cutting larger KR crystals as in [17].
6. The KR
2. Quantum affine algebra and crystal
2.1. Affine algebra E(1)
E(1)
6. The Dynkin diagram is depicted in Figure 1. Note that we follow [11] for
the labeling of the Dynkin nodes. It is different from that in [1] or [5]. Let I
be the index set of the Dynkin nodes, and let αi,α∨
simple coroots, fundamental weights, respectively. Following the notation in [11]
we denote the projection of Λionto the weight space of E6by Λi(i ∈ I0) and set
P =?
the Dynkin diagram of E(1)
6.
6. We consider in this paper the exceptional affine algebra
i,Λi(i ∈ I) be simple roots,
i∈I0ZΛi,P
+=?
i∈I0Z≥0Λi. Let (Cij)i,j∈Istand for the Cartan matrix for
6. For i,j ∈ I, i ∼ j means Cij= −1, namely, the nodes i and j are adjacent in
E(1)
2.2. KR crystal. Let g be any affine algebra and U′
tized enveloping algebra without the degree operator. Among finite-dimensional
U′
q(g)-modules there is a distinguished family called Kirillov-Reshetikhin (KR) mod-
ules [18, 24, 10]. One of the remarkable properties of KR modules is the existence
of a crystal basis [14] called a KR crystal. It was conjectured in [8, 7], and recently
settled for all nonexceptional types in [27]. The KR crystal is indexed by (a,i)
(a ∈ I0,i ∈ Z>0) and denoted by Ba,i. For exceptional types the KR crystal is
2
q(g) the corresponding quan-

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Figure 2. Crystal graph for B1,1
known to exist when the KR module is irreducible or the index a is adjacent to
0 [13]. Recently, the explicit crystal structure of all such cases of type E(1)
clarified in [5].
The KR crystal we are interested in in this paper is an E(1)
crystal structure was clarified in [5]. The crystal structure of B1,1is depicted in
Figure 2. Here vertices in the graph signify elements of B1,1and b
fib = b′or equivalently b = eib′. We adopt the original convention for the tensor
product of crystals. Namely, if B1and B2are crystals, then for b1⊗ b2∈ B1⊗ B2
the action of eiis defined as
?
b1⊗ eib2
6
was
6-crystal B1,1, whose
i
−→ b′stands for
ei(b1⊗ b2) =
eib1⊗ b2
if ϕi(b1) ≥ εi(b2),
else,
where εi(b) = max{k | ek
By glancing at Figure 2, one obtains the following lemma which will be used to
prove our main theorem. Let B0be the subgraph obtained by ignoring the 0-arrows
3
ib ?= 0} and ϕi(b) = max{k | fk
ib ?= 0}.

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from B. A route is a sequence (β1,...,βl) of arrows such that the sink of βjis the
source of βj+1for j = 1,...,l − 1.
Lemma 2.1. The graph B0has the following features.
(1) Suppose the initial arrow of a route R has the same color a as the terminal
arrow and there is no intermidiate arrow of color a. Then there are exactly
two arrows βi(i = 1,2) of color bisuch that bi∼ a in R.
(2) Let R be a route starting from???? ????1, (a1,...,al) the colors from the initial
arrow to the terminal one in R. Then we have
l−1
?
(3) Let R be a route of two steps with colors (a,b) such that b ?∼ a. Then there
exists a route R′with colors (b,a) starting and terminating at the same
vertices as R.
(4) Let R be a route of colors (a1,...,al). Let vibe the source of the arrow of
color ai(i = 1,...,l). Suppose a1∼ aland ai?∼ alfor any i = 2,...,l−1.
Then there is an arrow of color alstarting from vifor any i = 2,...,l −1.
j=1
Cajal= δal,1− 1.
Proof. (1) and (3) can be checked by direct observations. (2) and (4) are derived
from (1) and (3).
?
In what follows in this paper we assume B = B1,1. The set of classically re-
stricted paths in B⊗Lof weight λ ∈ P
P(λ,L) = {b ∈ B⊗L| wt(b) = λ and eib = 0 for all i ∈ I0}.
One may check that the following are equivalent for b = b1⊗ b2⊗ ··· ⊗ bL∈ B⊗L
and λ ∈ P
(1) b is a classically restricted path of weight λ ∈ P
(2) b1⊗ ··· ⊗ bL−1 is a classically restricted path of weight λ − wt(bL), and
εi(bL) ≤ ?λ − wt(bL),α∨
The weight function wt : B → P is given by wt(b) =?
Example 2.2. The element
b =???? ????1 ·???? ????2 ·???? ????3 ·????
of B⊗6is a classically restricted path of weight Λ3. The dot · signifies ⊗.
+is by definition
(2.1)
+.
+.
i? for all i ∈ I0.
i∈I(ϕi(b)−εi(b))Λi. The
j=1wt(bj).
weight function wt : B⊗L→ P is defined by wt(b1⊗ ··· ⊗ bL) =?L
????
16 ·???? ????2 ·???? ????
24
2.3. One-dimensional sums. The energy function D : B⊗L→ Z gives the grad-
ing on B⊗L. In our case where a path is an element of the tensor product of a single
KR crystal it takes a simple form. Due to the existence of the universal R-matrix
and the fact that B⊗B is connected, by [12] there is a unique (up to global additive
constant) function H : B ⊗ B → Z called the local energy function, such that


We normalize H by the condition
(2.3)
(2.2)
H(ei(b ⊗ b′)) = H(b ⊗ b′) +



1
−1
0
if i = 0 and e0(b ⊗ b′) = e0b ⊗ b′
if i = 0 and e0(b ⊗ b′) = b ⊗ e0b′
otherwise.
H(???? ????1 ⊗???? ????1) = 0.
4

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More specifically, the value of H is calculated as follows. Firstly, one knows the
crystal graph of B0⊗ B0decomposes into three connected components as
B0⊗ B0= B(2Λ1) ⊕ B(Λ1+ Λ2) ⊕ B(Λ1+ Λ5),
where B(λ) stands for the highest weight E6-crystal of highest weight λ and the
highest weight vector of each component is given by???? ????1 ⊗???? ????1,???? ????1 ⊗???? ????2,???? ????1 ⊗????
constant on each component, and takes the value 0,−1,−2, respectively. One can
confirm it from the fact that e0(???? ????1 ⊗???? ????1) =???? ????1 ⊗????
to the second and third component.
With this H the energy function D is defined by
????
????
18. H is
????
17 and e0(???? ????1 ⊗???? ????2) =???? ????1 ⊗????
22 belong
(2.4)
D(b1⊗ ··· ⊗ bL) =
L−1
?
j=1
(L − j) H(bj⊗ bj+1).
Define the one-dimensional sum X(λ,L;q) ∈ Z≥0[q−1] by
(2.5)
X(λ,L;q) =
?
b∈P(λ,L)
qD(b).
3. Rigged configuration and the bijection
3.1. The fermionic formula. This subsection reviews the definition of the fermionic
formula from [7, 8]. We at first provide the definition that is valid for any simply-
laced affine type g and datum L, and then restrict g and L to E(1)
corresponding to paths we consider in this paper.
L = (L(a)
another such matrix. Say that ν is an admissible configuration if it satisfies
?
i∈Z>0
i∈Z>0
6
+and a matrix
and the case
Fix λ ∈ P
i)a∈I0,i∈Z>0of nonnegative integers, almost all zero. Let ν = (m(a)
i) be
(3.1)
a∈I0
im(a)
iαa=
?
a∈I0
iL(a)
iΛa− λ
and
(3.2)
p(a)
i
≥ 0for all a ∈ I0and i ∈ Z>0,
where
(3.3)
p(a)
i
=
?
j∈Z>0
?
L(a)
j
min(i,j) −
?
b∈I0
(αa|αb)min(i,j)m(b)
j
?
.
Write C(λ,L) for the set of admissible configurations for λ ∈ P
the charge of a configuration ν by
c(ν) =1
2
a,b∈I0
j,k∈Z>0
(3.4)
+and L. Define
??
?
(αa|αb)min(j,k)m(a)
jm(b)
k
−
a∈I0
?
j,k∈Z>0
min(j,k)L(a)
jm(a)
k.
Using (3.3) c(ν) is rewritten as
(3.5)
c(ν) = −1
2


?
a∈I0,i∈Z>0
p(a)
im(a)
i
+
?
a∈I0,j,k∈Z>0
min(j,k)L(a)
jm(a)
k

.
5