Mark Shimozono

Affine Type A Crystal Structure on Tensor Products of Rectangles, Demazure Characters, and Nilpotent Varieties

Answering a question of Kuniba, Misra, Okado, Takagi, and Uchiyama, it is shown that certain higher level Demazure characters of affine type A, coincide with the graded characters of coordinate rings of closures of conjugacy classes of nilpotent matrices.

Graded Characters of Modules Supported in the Closure of a Nilpotent Conjugacy Class

This is a combinatorial study of the Poincare polynomials of isotypic components of a natural family of graded GL(n)-modules supported in the closure of a nilpotent conjugacy class. These polynomials generalize the Kostka?Foulkes polynomials and are q -analogues of Littlewood?Richardson coefficients. The coefficients of two-column Macdonald?Kostka polynomials also occur as a special case. It is conjectured that these q -analogues are the generating function of so-called catabolizable tableaux with the charge statistic of Lascoux and Schutzenberger. A general approach for a proof is given, and is completed in certain special cases including the Kostka?Foulkes case. Catabolizable tableaux are used to prove a characterization of Lascoux and Schutzenberger for the image of the tableaux of a given content under the standardization map that preserves the cyclage poset.

Fermionic Formulas for Level-Restricted Generalized Kostka Polynomials and Coset Branching Functions

Abstract: Level-restricted paths play an important rôle in crystal theory. They correspond to certain highest weight vectors of modules of quantum affine algebras. We show that the recently established bijection between Littlewood–Richardson tableaux and rigged configurations is well-behaved with respect to level-restriction and give an explicit characterization of level-restricted rigged configurations. As a consequence a new general fermionic formula for the level-restricted generalized Kostka polynomial is obtained. Some coset branching functions of type A are computed by taking limits of these fermionic formulas.

Key polynomials and a flagged Littlewood-Richardson rule

Abstract This paper studies a family of polynomials called key polynomials , introduced by Demazure and investigated combinatorially by Lascoux and Schutzenberger. We give two new combinatorial interpretations for these key polynomials and show how they provide the connection between two relatively recent combinatorial expressions for Schubert polynomials . We also give a flagged Littlewood—Richardson rule , an expansion of a flagged skew Schur function as a nonnegative sum of key polynomials.

X = M FOR SYMMETRIC POWERS

Author(s): Schilling, Anne; Shimozono, Mark | Abstract: The X=M conjecture of Hatayama et al. asserts the equality between the one-dimensional configuration sum X expressed as the generating function of crystal paths with energy statistics and the fermionic formula M for all affine Kac--Moody algebra. In this paper we prove the X=M conjecture for tensor products of Kirillov--Reshetikhin crystals B^{1,s} associated to symmetric powers for all nonexceptional affine algebras.

A Generalization of the Kostka–Foulkes Polynomials

Combinatorial objects called rigged configurations give rise to q-analogues of certain Littlewood–Richardson coefficients. The Kostka–Foulkes polynomials and two-column Macdonald–Kostka polynomials occur as special cases. Conjecturally these polynomials coincide with the Poincaré polynomials of isotypic components of certain graded GL(n)-modules supported in a nilpotent conjugacy class closure in gl(n).

A Cyclage Poset Structure for Littlewood-Richardson Tableaux

A graded poset structure is defined for sets of Littlewood?Richardson (LR) tableaux whose cardinalities are the multiplicities of irreducible gl(n)-modules in the tensor product of several irreducible gl(n)-modules indexed by rectangular partitions. This is a generalization of the cyclage poset on tableaux defined by Lascoux and Schutzenberger. This combinatorial construction is of independent interest and may be understood without reference to the underlying algebraic geometry. It is shown that the polynomials obtained by enumerating LR tableaux by shape and a generalized charge statistic, are the graded multiplicities of irreducibles in certain graded gl(n)-modules supported in the closure of a nilpotent conjugacy class. In particular, explicit tableau formulas are obtained for the special cases of the Kostka?Foulkes polynomials, the coefficient polynomials of two-column Macdonald?Kostka polynomials, and the graded multiplicities of coordinate rings of closures of conjugacy classes of nilpotent matrices. These polynomials coincide with the q -enumeration of rigged configurations and conjecturally coincide with the q -analogues of LR coefficients defined by the spin?weight generating functions of ribbon tableaux introduced by Lascoux, Leclerc, and Thibon.

A Uniform Model for Kirillov–Reshetikhin Crystals I: Lifting the Parabolic Quantum Bruhat Graph

© 2014

Virtual crystals and Kleber's algorithm

Abstract: Kirillov and Reshetikhin conjectured what is now known as the fermionic formula for the decomposition of tensor products of certain finite dimensional modules over quantum affine algebras. This formula can also be extended to the case of q-deformations of tensor product multiplicities as recently conjectured by Hatayama et al. In its original formulation it is difficult to compute the fermionic formula efficiently. Kleber found an algorithm for the simply-laced algebras which overcomes this problem. We present a method which reduces all other cases to the simply-laced case using embeddings of affine algebras. This is the fermionic analogue of the virtual crystal construction by the authors, which is the realization of crystal graphs for arbitrary quantum affine algebras in terms of those of simply-laced type.

Virtual crystals and fermionic formulas of type _{+1}⁽²⁾, _{2}⁽²⁾, and _{}⁽¹⁾

A device having utility in interfering with a persons desire to hold an object such as a cigarette, cigar or pipe, between his lips, i.e., an anti-smoking device. In the form of a cigarette holder, it includes a generally tubular shell having first and second ends, with the first end being adapted to receive a cigarette; the second end thereof includes structure adapted to be held between a persons lips. A DC voltage source (such as a dry cell battery) of at least six volts and preferably nine volts is mounted within the shell. First and second electrically conductive members are connected to the output of the DC source, with the distal ends of said conductive members extending alongside the lip-contacting structure so that they may be readily touched by a persons lips. The distal ends of said conductive members are separated so as to form a normally open electrical path, such that placing the lip-contacting structure between a persons lips will instantaneously close the electrical path and result in the discharge of DC current from said source through the lips. A potentiometer is optionally provided to adjust the flow of current from a minimum of about one milliamp (in order to be discernable) to a maximum of about five milliamps (so as to avoid intolerable sensations). Additionally, means are disclosed for re-charging a battery which is permanently mounted within the shell of a cigarette holder or the like.