An iterative formula for the Kostka–Foulkes polynomials

Journal of Algebraic Combinatorics - An iterative formula for the Kostka–Foulkes polynomials is given using the vertex operator realization of the Hall–Littlewood polynomials. The...     

The Bailey Lemma and Kostka Polynomials

Using the theory of Kostka polynomials, we prove an A _n–1 version of Bailey's lemma at integral level. Exploiting a new, conjectural expansion for Kostka numbers, this is then generalized to fractional levels, leading to a new expression for admissible characters of A⁽¹⁾ _n–1 and to identities for A-type branching functions.     

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).     

On the Connection Between Macdonald Polynomials and Demazure Characters

We show that the specialization of nonsymmetric Macdonald polynomials at t = 0 are, up to multiplication by a simple factor, characters of Demazure modules for . This connection furnishes Lie-theoretic proofs of the nonnegativity and monotonicity of Kostka polynomials.     

EXOTIC SYMMETRIC SPACE OVER A FINITE FIELD,II

This paper is the second of the papers of the same title. In this paper, we prove a conjecture of Achar–Henderson, which asserts that the Poincaré polynomials of the intersection cohomology complex associated to the closure of Sp_2n -orbits in the Kato's exotic nilpotent cone coincide with the modified Kostka polynomials indexed by double partitions, introduced by the first author. Actually, this conjecture was recently proved by Kato by a different method. Our approach is based on the theory of character sheaves on the exotic symmetric space.     

Quasisymmetric Schur functions

We introduce a new basis for quasisymmetric functions, which arise from a specialization of nonsymmetric Macdonald polynomials to standard bases, also known as Demazure atoms. Our new basis is called the basis of quasisymmetric Schur functions, since the basis elements refine Schur functions in a natural way. We derive expansions for quasisymmetric Schur functions in terms of monomial and fundamental quasisymmetric functions, which give rise to quasisymmetric refinements of Kostka numbers and standard (reverse) tableaux. From here we derive a Pieri rule for quasisymmetric Schur functions that naturally refines the Pieri rule for Schur functions. After surveying combinatorial formulas for Macdonald polynomials, including an expansion of Macdonald polynomials into fundamental quasisymmetric functions, we show how some of our results can be extended to include the t parameter from Hall-Littlewood theory.     

The t-analog of the level one string function for twisted affine Kac–Moody algebras

We study Lusztig?s t-analog of weight multiplicities, or affine Kostka–Foulkes polynomials, associated to level one representations of twisted affine Kac–Moody algebras. We obtain an explicit closed form expression for the unique t-string function, using constant term identities of Macdonald and Cherednik. This extends previous work on t-string functions for the untwisted simply-laced affine Kac–Moody algebras.     

Branching formula for q-Littlewood–Richardson coefficients

A generalized inversion statistic is introduced on k-tuples of semistandard tableaux. It is shown that the cospin of a semistandard k-ribbon tableau is equal to the generalized inversion number of its k-quotient. This leads to a branching formula for the q-analogue of Littlewood–Richardson coefficients defined by Lascoux, Leclerc, and Thibon. This branching formula generalizes a recurrence of Garsia and Procesi involving Kostka–Foulkes polynomials.     

Degrees of stretched Kostka coefficients

Given a partition λ and a composition β, the stretched Kostka coefficient is the map n ↦ K _{n λ,n β} sending each positive integer n to the Kostka coefficient indexed by n λ and n β. Kirillov and Reshetikhin (J. Soviet Math. 41(2), 925–955, 1988) have shown that stretched Kostka coefficients are polynomial functions of n. King, Tollu, and Toumazet have conjectured that these polynomials always have nonnegative coefficients (CRM Proc. Lecture Notes 34, 99–112, 2004), and they have given a conjectural expression for their degrees (Séminaire Lotharingien de Combinatoire 54A, 2006). We prove the values conjectured by King, Tollu, and Toumazet for the degrees of stretched Kostka coefficients. Our proof depends upon the polyhedral geometry of Gelfand–Tsetlin polytopes and uses tilings of GT-patterns, a combinatorial structure introduced in De Loera and McAllister, (Discret. Comput. Geom. 32(4), 459–470, 2004). Research supported by NSF VIGRE Grant No. DMS-0135345 and by NWO Mathematics Cluster DIAMANT.     

Recurrence Formulas for Kostka and Inverse Kostka Numbers via Quantum Cohomology of Grassmannians

A Gröbner basis for the small quantum cohomology of Grassmannian G _k,n is constructed and used to obtain new recurrence relations for Kostka numbers and inverse Kostka numbers. Using these relations it is shown how to determine inverse Kostka numbers which are related to the mod-p Wu formula.     

A bijection between Littlewood-Richardson tableaux and rigged configurations

We define a bijection from Littlewood-Richardson tableaux to rigged configurations and show that it preserves the appropriate statistics. This proves in particular a quasi-particle expression for the generalized Kostka polynomials labeled by a partition and a sequence of rectangles R. The generalized Kostka polynomials are q-analogues of multiplicities of the irreducible -module of highest weight in the tensor product .     

New fermionic formula for unrestricted Kostka polynomials

A new fermionic formula for the unrestricted Kostka polynomials of type is presented. This formula is different from the one given by Hatayama et al. and is valid for all crystal paths based on Kirillov-Reshetikhin modules, not just for the symmetric and antisymmetric case. The fermionic formula can be interpreted in terms of a new set of unrestricted rigged configurations. For the proof a statistics preserving bijection from this new set of unrestricted rigged configurations to the set of unrestricted crystal paths is given which generalizes a bijection of Kirillov and Reshetikhin.     

Generalization of the Gale–Ryser Theorem

We prove an inequality for the Kostka–Foulkes polynomials K_λ,μ(q) and give a criteria for the existence of a unique configuration of the given type (λ, μ). As a corollary, we obtain a nontrivial lower bound for the Kostka numbers which is a generalization the Gale–Ryser theorem on an existence of a (0,1)-matrix with given sums of rows and columns. A new proof of the Berenstein–Zelevinsky weight-multiplicity-one criteria is given.     

Recurrences for characters of the symmetric group

Recurrences for irreducible and Kostka characters of the symmetric group are derived here in a purely combinatorial treatment.     

Kostka functions associated to complex reflection groups and a conjecture of Finkelberg-Ionov

Kostka functions K_(λ,μ)~±(t), indexed by r-partitions λ and μ of n, are a generalization of Kostka polynomials K_(λ,μ)(t) indexed by partitions λ,μ of n. It is known that Kostka polynomials have an interpretation in terms of Lusztig's partition function. Finkelberg and Ionov(2016) defined alternate functions K_(λ,μ)(t) by using an analogue of Lusztig's partition function, and showed that K_(λ,μ)(t) ∈Z≥0[t] for generic μ by making use of a coherent realization. They conjectured that K_(λ,μ)(t) coincide with K_(λ,μ)~-(t). In this paper, we show that their conjecture holds. We also discuss the multi-variable version, namely, r-variable Kostka functions K_(λ,μ)~±(t_1,…,t_r).     

Higher spin generalisation of Fueter's theorem

In this paper, we generalize Fueter's theorem to the higher spin setting. To do so, we consider an alternative proof for the celebrated theorem that uses the Fischer decomposition. This decomposition is then extended to spaces of polynomials that depend on wedge variables, after which we can finish the proof of our higher spin Fueter theorem.     

On a Formula for the Kostka Numbers

From Kostant’s multiplicity formula for general linear groups, one can derive a formula for the Kostka numbers. In this note we give a combinatorial proof of this formula. Received January 7, 2005     

A combinatorial interpretation of the inverse kostka matrix

The Kostka matrix K relates the. homogeneous and the Schur bases in the ring of symmetric functions where K _λ,μenumerates the number of column strict tableaux of shape λ and type μ. We make use of the Jacobi -Trudi identity to give a combinatorial interpretation for the inverse of the Kostka matrix in terms of certain types of signed rim hook tabloids. Using this interpretation, the matrix identity KK ^?1=Iis given a purely combinatorial proof. The generalized Jacobi-Trudi identity itself is also shown to admit a combinatorial proof via these rim hook tabloids. A further application of our combinatorial interpretation is a simple rule for the evaluation of a specialization of skew Schur functions that arises in the computation of plethysms.     

A combinatorial interpretation of the inverse kostka matrix

The Kostka matrix K relates the. homogeneous and the Schur bases in the ring of symmetric functions where K_λ,μenumerates the number of column strict tableaux of shape λ and type μ. We make use of the Jacobi -Trudi identity to give a combinatorial interpretation for the inverse of the Kostka matrix in terms of certain types of signed rim hook tabloids. Using this interpretation, the matrix identity KK^-1=Iis given a purely combinatorial proof. The generalized Jacobi-Trudi identity itself is also shown to admit a combinatorial proof via these rim hook tabloids. A further application of our combinatorial interpretation is a simple rule for the evaluation of a specialization of skew Schur functions that arises in the computation of plethysms.