Affine crystal structure on rigged configurations of type D_n^(1)

Abstract

Extending the work arXiv:math/0508107, we introduce the affine crystal action
on rigged configurations which is isomorphic to the Kirillov-Reshetikhin
crystal B^{r,s} of type D_n^(1) for any r,s. We also introduce a representation
of B^{r,s} (r not equal to n-1,n) in terms of tableaux of rectangular shape r x
s, which we coin Kirillov-Reshetikhin tableaux (using a non-trivial analogue of
the type A column splitting procedure) to construct a bijection between
elements of a tensor product of Kirillov-Reshetikhin crystals and rigged
configurations.

Full text

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arXiv:1109.3523v1 [math.QA] 16 Sep 2011
AFFINE CRYSTAL STRUCTURE ON RIGGED
CONFIGURATIONS OF TYPE D(1)
n
MASATO OKADO, REIHO SAKAMOTO, AND ANNE SCHILLING
Abstract. Extending the work [24], we introduce the affine crystal action on
rigged configurations which is isomorphic to the Kirillov–Reshetikhin crystal
Br,sof type D(1)
n
for any r,s. We also introduce a representation of Br,s
(r ?= n−1,n) using tableaux of rectangular shape r×s, which are obtained from
a column splitting procedure that was effectively used for type A to construct
a bijection from elements of a tensor product of Kirillov–Reshetikhin crystals
to rigged configurations.
Contents
1.
Acknowledgments
2. Background on crystals
2.1.Review of crystals and notation
2.2.
±-diagrams and definition of σ
2.3.Kirillov–Reshetikhin crystals of type D(1)
3.Rigged configurations
4.Affine crystal structure on rigged configurations
4.1.
±-diagrams on rigged configurations
4.2.Affine crystal structure
5.Combinatorial bijection
5.1.The filling map
5.2.Operations on the rigged configurations
5.3.The combinatorial bijection
5.4.Conjectures and open questions
References
Introduction
1
3
3
3
3
4
5
8
8
n
13
14
15
16
19
21
23
1. Introduction
Motivated by studies using the Bethe Ansatz, Kerov, Kirillov and Reshetikhin [9]
introduced interesting new combinatorialobjects coined rigged configurations (RCs),
and found a bijection to semistandard tableaux. It was later realized [10] that this
type of bijection can be extended to a bijection between RCs and elements in a
multiple tensor product of Kirillov–Reshetikhin crystals of type A satisfying high-
est weight conditions. Since there is an action of Kashiwara operators on crystals, it
1991 Mathematics Subject Classification. Primary 17B37; Secondary: 05A19; 05A15; 81R50;
82B23.
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2M. OKADO, R. SAKAMOTO, AND A. SCHILLING
is natural to try to find the action on RCs through the bijection. This was achieved
by the third author in [24] for fi,eiwith i ?= 0, and subsequently in [27] for i = 0
by considering the action of the so-called promotion operator on RCs.
We wish to consider a similar problem for type D(1)
of irreducible components in a multiple tensor product turns out large, we start
by considering the single KR crystal Br,sin this paper. For classical Kashiwara
operators fi,ei(i ?= 0) the action has already been provided in [24]. The action of
f0and e0is defined by
n . Since the multiplicities
f0= σ ◦ f1◦ σande0= σ ◦ e1◦ σ
using the involution σ corresponding to exchanging the Dynkin nodes 0 and 1. Since
σ commutes with fi,eifor i = 2,...,n, we are left to define the action of σ on RCs
which are {2,...,n}-highest weight. This is performed by defining a map γrcfrom
±-diagrams, another combinatorial object parameterizing {2,...,n}-highest weight
elements, to {2,...,n}-highest weight RCs. This map has an interesting feature.
We cut a given ±-diagram into columns, associate to each column an atomic RC,
and add up these atomic RCs in a certain way. In other words, γrc can be in-
terpreted as a linear map. Such a nice relationship between rigged configurations
and ±-diagrams was originally suggested through the analysis of the expression of
the combinatorial R-matrix [18] written in terms of ±-diagrams via the expected
property that the combinatorial R-matrix act as the identity on the rigged config-
urations (Conjecture 5.11). Note that the original bijection between ±-diagrams
and {2,3,...,n}-highest weight Kashiwara–Nakashima tableaux [8] discovered by
the third author [25] requires some non-trivial algorithm.
Combining these results we obtain our first main theorem (Theorem 4.9) describ-
ing the affine crystal structure on rigged configurations corresponding to a single
Kirillov–Reshetikhin crystal of type D(1)
n . The trivial action of the combinatorial
R-matrix on the level of rigged configurations makes it desirable to find an explicit
description of the bijection Φ between rigged configurations and elements of tensor
product of Kirillov–Reshetikhin crystals.
This leads us to the next purpose of the present paper. We provide an ex-
plicit combinatorial algorithm for Φ (Section 5), which leads to a completely new
set of tableaux. For type A the bijection from rigged configurations to Kirillov–
Reshetikhin crystals is given by successively applying a fundamental algorithm δ.
Each application of δ produces a letter, which can be placed in the r × s rectangle
corresponding to the Kirillov–Reshetikhin crystal Br,sresulting into a semistan-
dard tableau. The algorithm for δ also exists for type D [20] for tensor products
of the Kirillov–Reshetikhin crystal associated to the vector representation. We ex-
tend this to arbitrary Kirillov–Reshetikhin crystals Br,s, which produces tableaux
whose shape is an r × s rectangle. We note that these tableaux are completely
different from the usual Kashiwara–Nakashima tableaux [8], since in their repre-
sentation tableaux do not necessarily have rectangular shape. The new tableaux
can be thought of as Kirillov–Reshetikhin tableaux. The correspondence between
Kashiwara–Nakashima and Kirillov–Reshetikhin tableaux is called the filling map
(Section 5.1). At the end of the paper we give several conjectures (Conjectures 5.10,
5.11, and 5.15) that provide evidence that our filling map is natural and useful.
The paper is organized as follows. In Section 2 we review facts about crystal
bases that will be needed for this paper. Rigged configurations and background

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AFFINE CRYSTAL STRUCTURE ON RIGGED CONFIGURATIONS OF TYPE D(1)
n
3
material are presented in Section 3. Section 4 contains the main results, namely
the affine crystal structure on rigged configurations for a single tensor factor for
type D(1)
n . The combinatorial bijection and associated conjectures are the subject
of Section 5. It is shown in Theorem 5.9 that for the classically highest weight
elements in a single Kirillov–Reshetikhin crystal the combinatorial definition of the
bijection agrees with the correspondence under the affine crystal isomorphism.
Acknowledgments. MO is partially supported by the Grants-in-Aid for Scientific
Research No. 23340007 and No. 23654007 from JSPS. RS is partially supported
by Grants-in-Aid for Scientific Research No. 21740114 from JSPS. AS was in part
supported by NSF grants DMS–0652641 and DMS–1001256.
2. Background on crystals
In this section we review some facts needed about crystal bases.
2.1. Review of crystals and notation. Crystal theory was introduced by Kashi-
wara [7] and provides a combinatorial approach in terms of tableaux to the rep-
resentation theory of quantum groups and Lie algebras. A crystal is a nonempty
set B together with Kashiwara lowering and raising operators fiand eifor i ∈ I,
where I is the index set of the Dynkin diagram of the associated Lie algebra g. The
Kashiwara operators are q → 0 limits of the Chevalley operators of the correspond-
ing quantum algebra Uq(g). One of the amazing properties of crystals is the fact
that they are well-behaved with respect to tensor products. Given two g-crystals
B and B′, the Kashiwara operators on the tensor product B ⊗B′can be described
by a completely combinatorial rule called the signature rule. For an introduction
to crystal theory see for example the book by Hong and Kang [4].
For an affine Kac–Moody algebra g, we denote by g0the finite-dimensional simple
Lie algebra obtained by removing the 0 node from the Dynkin diagram of g and
by αi (i ∈ I) the simple roots. We also denote by ̟i (i ∈ I0 := I \ {0}) the
fundamental weights of g0. Let Λ be a dominant weight of g0. For crystals B(Λ)
associated to highest weight representations of highest weight Λ of Uq(g0), there
exist generalizations of the usual semistandard Young tableaux (which we can think
of as type A objects) known as Kashiwara–Nakashima (KN) tableaux [8]. For type
Dnthese are tableaux of shape Λ over some ordered alphabet {1 < 2 < ... < n,n <
... < 2 < 1}. Here the letters n and n are incomparable. For the precise definition
of the semistandard condition for type Dnsee [4]. Let B be a crystal of type g and
b ∈ B. For a subset J ⊂ I we say that b is J-highest weight if ei(b) = 0 for i ∈ J.
We say b is highest weight if it is I0-highest weight.
In this paper we freely identify dominant weights (without spin nodes) and parti-
tions. More precisely, given a partition λ = (λ1≥ λ2≥ ··· ≥ λℓ) with at most n−2
parts (that is ℓ ≤ n−2), we can associate the dominant weight Λ = ̟i1+···+̟ik
where the ij for 1 ≤ j ≤ k = λ1 are the heights of the columns in λ. When we
draw the Ferrers diagram for the partition λ with λiboxes in row i, we use English
notation adjusting the rows on the left and placing the largest part on the top. The
height of a cell in a partition is equal to its row index (that is the distance from
the top of the Ferrers diagram). We also use English convention for tableaux.
2.2. ±-diagrams and definition of σ. In order to define Kirillov–Reshetikhin
crystals for type D(1)
n
following [2, 25], we need to define an involution σ which

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4M. OKADO, R. SAKAMOTO, AND A. SCHILLING
corresponds to the type D(1)
0 and 1. This is achieved by noting that σ commutes with the Kashiwara crystal
operators fiand eifor i ∈ {2,3,...,n}. Then σ is defined explicitly on {2,3,...,n}-
highest weight vectors.
It turns out that {2,3,...,n}-highest weight vectors are in bijection with so-
called ±-diagrams (see Proposition 2.1 below). A ±-diagram P is a sequence of
shapes λ ⊂ µ ⊂ Λ such that Λ/µ and µ/λ are horizontal strips (i.e. every column
contains at most one box). We depict this ±-diagram by the skew tableau of shape
Λ/λ in which the cells of µ/λ are filled with the symbol + and those of Λ/µ are
filled with the symbol −. The partition Λ is called the outer shape of P and λ
is called the inner shape of P. In this paper we only require ±-diagrams for the
nonspin case, that is, when the height of Λ is at most n − 2.
For our purposes it will be convenient to state the bijection between ±-diagrams
and {2,3,...,n}-highest weight elements in an inductive fashion.
n Dynkin diagram automorphism of interchanging nodes
Proposition 2.1. [25] [2, Section 3.2] There is a bijection γ from ±-diagrams of
outer shape Λ to {2,3,...,n}-highest weight elements in the highest weight crystal
B(Λ). The ±-diagram P which has + in every column and no − corresponds to the
highest weight vector u ∈ B(Λ) of weight equal to the outer shape of P. Given a ±-
diagram P we can obtain the corresponding {2,3,...,n}-highest element γ(P) = b
inductively as follows:
Case 1: P has a column where a + can be added.
Let P′be the ±-diagram obtained from P by adding a + in the rightmost
possible column at height h. Then b = f1f2···fhγ(P′).
Case 2: P has no column where a + can be added and at least one −.
Let P′be the ±-diagram obtained from P by removing the leftmost − at
height h and either moving the + in the same column down if h > 1 or
adding a + if h = 1. Then b = f1f2···fnfn−2fn−3···fhγ(P′).
We now define the following map S on ±-diagrams. Let c.(h), c+(h), c−(h),
c±(h) be the number of columns in P of outer height h with no sign, +, −, ±,
respectively. As we will see in (2.1) below, in our setting the values of h are either
all even or all odd. Note that P is specified if all values c.(h), c+(h), c−(h), c±(h)
are given.
Definition 2.2. Let P be a ±-diagram of outer shape Λ, where the columns in
Λ are either all of even or all of odd height. Then S(P) is the ± diagram, where
compared to P the values c+(h) and c−(h) are interchanged for h ≥ 1, and the
values of c.(h − 2) and c±(h) are interchanged for h ≥ 2.
2.3. Kirillov–Reshetikhin crystals of type D(1)
Lie algebra with index set I = {0,1,...,n}. Kirillov–Reshetikhin (KR) crystals
Br,sare indexed by r ∈ I0 := I \ {0} and an integer s ≥ 1. For nonexceptional
types their existence was proven in [17, 19]. In this paper we only deal with the
KR crystals Br,sof type D(1)
n .
As classical crystals the Kirillov–Reshetikhin crystals usually decompose into
several components. For type D(1)
n − 2 is
Br,s∼=
B(λ)
n . Let g be an affine Kac–Moody
n , the classical decomposition of Br,sfor 1 ≤ r ≤
(2.1)
?
λ
as Dn-crystals,

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AFFINE CRYSTAL STRUCTURE ON RIGGED CONFIGURATIONS OF TYPE D(1)
n
5
where the sum is over all partitions (or equivalently weights) obtained from (sr)
by removing vertical dominoes. Each term appears with multiplicity one. For the
spin cases r = n − 1,n we have
(2.2)Br,s∼= B(s̟r)as Dn-crystals.
The affine Kashiwara crystal operators f0and e0are defined as
(2.3)f0= σ ◦ f1◦ σande0= σ ◦ e1◦ σ ,
where σ is the analogue of the Dynkin automorphism which interchanges nodes 0
and 1 as given in the next definition.
Definition 2.3. Let t ∈ Br,swith Br,sa KR crystal of type D(1)
n − 2. Choose a sequence b = (b1,b2,...,bk) with bi ∈ {2,3,...,n} such that
eb(t) := eb1eb2···ebk(t) is {2,3,...,n}-highest weight. Then define
n
with 1 ≤ r ≤
(2.4)σ(t) := fb′ ◦ γ ◦ S ◦ γ−1◦ eb(t) ,
where b′is the reverse of b so that fb′ := fbk···fb1, and S as in Definition 2.2
with the heights restricted to 1 ≤ h ≤ r.
Let B1, B2be two affine crystals with generators v1and v2, respectively, such
that B1⊗B2is connected and v1⊗v2lies in a one-dimensional weight space. By [15,
Proposition 3.8], this holds for any two KR crystals. The generator v for the KR
crystal Br,sis chosen to be the unique element of classical weight s̟r.
The combinatorial R-matrix [6, Section 4] is the unique affine crystal isomor-
phism
(2.5)R : B1⊗ B2→ B2⊗ B1.
By weight considerations, this must satisfy R(v1⊗ v2) = v2⊗ v1.
On (tensor products of) Kirillov–Reshetikhin crystals, there is a (co)energy
function defined. The (co)energy is constant on classical components. For a single
Br,sthe coenergy for the classical component B(λ) is equal to the number of vertical
dominoes in (sr)\λ [3] (see also [26, Definition 5.4, Theorem 7.5]). We denote the
coenergy function by
(2.6)
D : B → Z.
For the definition of the (co)energy on general tensor factors, see for example [21,
Theorem 2.4] [26].
3. Rigged configurations
Let g be a simply-laced affine Kac–Moody algebra with index set I of the under-
lying Dynkin diagram. Recall that I0= I \ {0} is the index set of the underlying
algebra of finite type g0and set H = I0×Z>0. The (highest-weight) rigged config-
urations are indexed by a multiplicity array L = (L(a)
integers and a dominant weight Λ of g0. Note that only finitely many L(a)
multiplicity array L are nonzero. The sequence of partitions ν = {ν(a)| a ∈ I0} is
an (L,Λ)-configuration if
?
i
| (a,i) ∈ H) of nonnegative
i
in the
(3.1)
(a,i)∈H
im(a)
iαa=
?
(a,i)∈H
iL(a)
i̟a− Λ,