103 research outputs found
Demazure crystals, Kirillov-Reshetikhin crystals, and the energy function
It has previously been shown that, at least for non-exceptional Kac-Moody Lie
algebras, there is a close connection between Demazure crystals and tensor
products of Kirillov-Reshetikhin crystals. In particular, certain Demazure
crystals are isomorphic as classical crystals to tensor products of
Kirillov-Reshetikhin crystals via a canonically chosen isomorphism. Here we
show that this isomorphism intertwines the natural affine grading on Demazure
crystals with a combinatorially defined energy function. As a consequence, we
obtain a formula of the Demazure character in terms of the energy function,
which has applications to Macdonald polynomials and q-deformed Whittaker
functions.Comment: 35 pages. v2: minor revisions, including several new examples and
reference
On higher level Kirillov--Reshetikhin crystals, Demazure crystals, and related uniform models
We show that a tensor product of nonexceptional type Kirillov--Reshetikhin
(KR) crystals is isomorphic to a direct sum of Demazure crystals; we do this in
the mixed level case and without the perfectness assumption, thus generalizing
a result of Naoi. We use this result to show that, given two tensor products of
such KR crystals with the same maximal weight, after removing certain
-arrows, the two connected components containing the minimal/maximal
elements are isomorphic. Based on the latter fact, we reduce a tensor product
of higher level perfect KR crystals to one of single-column KR crystals, which
allows us to use the uniform models available in the literature in the latter
case. We also use our results to give a combinatorial interpretation of the
Q-system relations. Our results are conjectured to extend to the exceptional
types.Comment: 15 pages, 1 figure; v2, incorporated changes from refere
A uniform model for Kirillov-Reshetikhin crystals. Extended abstract
We present a uniform construction of tensor products of one-column
Kirillov-Reshetikhin (KR) crystals in all untwisted affine types, which uses a
generalization of the Lakshmibai-Seshadri paths (in the theory of the
Littelmann path model). This generalization is based on the graph on parabolic
cosets of a Weyl group known as the parabolic quantum Bruhat graph. A related
model is the so-called quantum alcove model. The proof is based on two lifts of
the parabolic quantum Bruhat graph: to the Bruhat order on the affine Weyl
group and to Littelmann's poset on level-zero weights. Our construction leads
to a simple calculation of the energy function. It also implies the equality
between a Macdonald polynomial specialized at t=0 and the graded character of a
tensor product of KR modules.Comment: 10 pages, 1 figur
Kirillov-Reshetikhin crystals for using Nakajima monomials
We give a realization of the Kirillov--Reshetikhin crystal using
Nakajima monomials for using the crystal structure
given by Kashiwara. We describe the tensor product in terms of a shift of indices, allowing us to recover the Kyoto
path model. Additionally, we give a model for the KR crystals using
Nakajima monomials.Comment: 24 pages, 6 figures; v2 improved introduction, added more figures,
and other misc improvements; v3 changes from referee report
A uniform realization of the combinatorial -matrix
Kirillov-Reshetikhin crystals are colored directed graphs encoding the
structure of certain finite-dimensional representations of affine Lie algebras.
A tensor products of column shape Kirillov-Reshetikhin crystals has recently
been realized in a uniform way, for all untwisted affine types, in terms of the
quantum alcove model. We enhance this model by using it to give a uniform
realization of the combinatorial -matrix, i.e., the unique affine crystal
isomorphism permuting factors in a tensor product of KR crystals. In other
words, we are generalizing to all Lie types Sch\"utzenberger's sliding game
(jeu de taquin) for Young tableaux, which realizes the combinatorial -matrix
in type . Our construction is in terms of certain combinatorial moves,
called quantum Yang-Baxter moves, which are explicitly described by reduction
to the rank 2 root systems. We also show that the quantum alcove model does not
depend on the choice of a sequence of alcoves joining the fundamental one to a
translation of it.Comment: arXiv admin note: text overlap with arXiv:1112.221
Uniform description of the rigged configuration bijection
We give a uniform description of the bijection from rigged
configurations to tensor products of Kirillov--Reshetikhin crystals of the form
in dual untwisted types: simply-laced types and
types , , , and . We give
a uniform proof that is a bijection and preserves statistics. We
describe uniformly using virtual crystals for all remaining types, but
our proofs are type-specific. We also give a uniform proof that is a
bijection for when , for all , map to
under an automorphism of the Dynkin diagram. Furthermore, we give a
description of the Kirillov--Reshetikhin crystals using tableaux of a
fixed height depending on in all affine types. Additionally, we are
able to describe crystals using shaped tableaux that
are conjecturally the crystal basis for Kirillov--Reshetikhin modules for
various nodes .Comment: 60 pages, 5 figures, 3 tables; v2 incorporated changes from refere
Crystal energy functions via the charge in types A and C
The Ram-Yip formula for Macdonald polynomials (at t=0) provides a statistic
which we call charge. In types A and C it can be defined on tensor products of
Kashiwara-Nakashima single column crystals. In this paper we prove that the
charge is equal to the (negative of the) energy function on affine crystals.
The algorithm for computing charge is much simpler and can be more efficiently
computed than the recursive definition of energy in terms of the combinatorial
R-matrix.Comment: 25 pages; 1 figur
Simplicity and similarity of Kirillov-Reshetikhin crystals
We show that the Kirillov-Reshetikhin crystal B^{r,s} for nonexceptional
affine types is simple and have the similarity property. As a corollary of the
first fact we can derive that the tensor product of KR crystals is connected.
Variations of the second property are also given.Comment: 11 pages. arXiv admin note: text overlap with arXiv:0810.506
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