664 research outputs found
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
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
-Schur functions and affine Schubert calculus
This book is an exposition of the current state of research of affine
Schubert calculus and -Schur functions. This text is based on a series of
lectures given at a workshop titled "Affine Schubert Calculus" that took place
in July 2010 at the Fields Institute in Toronto, Ontario. The story of this
research is told in three parts: 1. Primer on -Schur Functions 2. Stanley
symmetric functions and Peterson algebras 3. Affine Schubert calculusComment: 213 pages; conference website:
http://www.fields.utoronto.ca/programs/scientific/10-11/schubert/, updates
and corrections since v1. This material is based upon work supported by the
National Science Foundation under Grant No. DMS-065264
Generalized Weyl modules for twisted current algebras
We introduce the notion of generalized Weyl modules for twisted current
algebras. We study their representation-theoretic and combinatorial properties
and connection to the theory of nonsymmetric Macdonald polynomials. As an
application we compute the dimension of the classical Weyl modules in the
remaining unknown case.Comment: 24 pages, 2 figure
BGG reciprocity for current algebras
It was conjectured by Bennett, Chari, and Manning that a BGG-type reciprocity
holds for the category of graded representations with finite-dimensional graded
components for the current algebra associated to a simple Lie algebra. We
associate a current algebra to any indecomposable affine Lie algebra and show
that, in this generality, the BGG reciprocity is true for the corresponding
category of representations.Comment: 23 pg, corrections to Lemma 2.1
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