arXiv subjects

Mark Shimozono

Publications and source records attributed to Mark Shimozono.

63 records · Page 4Linked to original sources

arXiv2000

Hall-Littlewood vertex operators and generalized Kostka polynomials

A family of vertex operators that generalizes those given by Jing for the Hall-Littlewood symmetric functions is presented. These operators produce symmetric functions related to the Poincare polynomials referred to as generalized Kostka polynomials in the same way that Jing's operator produces symmetric functions related to Kostka-Foulkes polynomials. These operators are then used to derive commutation relations and new relations involving the generalized Kostka coefficients. Such relations may be interpreted as identities in the (GL(n) x C^*)-equivariant K-theory of the nullcone.

math.QA↗

arXiv2000

Fermionic formulas for level-restricted generalized Kostka polynomials and coset branching functions

Level-restricted paths play an important role 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.

math.QA↗

arXiv1999

Bosonic formula for level-restricted paths

We prove a bosonic formula for the generating function of level-restricted paths for the infinite families of affine Kac-Moody algebras. In affine type A this yields an expression for the level-restricted generalized Kostka polynomials.

math.QA↗

arXiv1999

A bijection between Littlewood-Richardson tableaux and rigged configurations

A bijection is defined from Littlewood-Richardson tableaux to rigged configurations. It is shown that this map preserves the appropriate statistics, thereby proving a quasi-particle expression for the generalized Kostka polynomials, which are q-analogues of multiplicities in tensor products of irreducible general linear group modules indexed by rectangular partitions.

math.CO↗

arXiv1998

Graded characters of modules supported in the closure of a nilpotent conjugacy class

We study 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 and are q-analogues of Littlewood-Richardson coefficients corresponding to arbitrary tensor products of irreducibles. Many properties and formulas for these polynomials are derived, such as a generalized Morris recurrence, q-Kostant formula, and a conjectural formula in terms of catabolizable tableaux and charge.

math.QA↗

arXiv1998

A cyclage poset structure for Littlewood-Richardson tableaux

A graded poset structure is defined for the sets of Littlewood-Richardson (LR) tableaux that count the multiplicity of an irreducible GL(n)-module in the tensor product of irreducibles indexed by a sequence of rectangular partitions. This poset generalizes the cyclage poset on column-strict tableaux defined by Lascoux and Schutzenberger, and its grading function generalizes the charge statistic. It is shown that the polynomials obtained by enumerating LR tableaux by shape and the generalized charge, are the Poincare polynomials of isotypic components of the certain modules supported in the closure of a nilpotent conjugacy class.

math.QA↗

arXiv1998

Multi-atoms and monotonicity of generalized Kostka polynomials

The generalized Kostka polynomials are the Poincare polynomials of isotypic components of certain graded GL(n)-modules. The former satisfy a monotonicity property arising from natural surjections of the corresponding modules. This monotonicity property is realized combinatorially by a directed system of order- and statistic-preserving embeddings between the sets of Littlewood-Richardson (LR) tableaux that describe the generalized Kostka polynomials. Each set of LR tableaux is embedded into a cocyclage poset of column-strict tableaux of a fixed content, and the image is characterized in terms of catabolizable tableaux.

math.QA↗

arXiv1998

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 Demazure characters of affine type A, coincide with the graded characters of coordinate rings of closures of conjugacy classes of nilpotent matrices. This entails a translation of the affine type A crystal theory into the language of tableaux following Nakayashiki and Yamada, for the case of tensor products of the classical crystals indexed by rectangular partitions. In particular the explicit action of the zero-th crystal raising operator on the above crystals is given, and its direct connection with the generalized cocyclage on Littlewood-Richardson tableaux is explained.

math.QA↗

arXiv1998

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 Poincare polynomials of isotypic components of certain graded GL(n)-modules supported in a nilpotent conjugacy class closure in gl(n).

math.QA↗