2021/10/28 by Cédric Lecouvey, Lecouvey, Cédric, Cristian Lenart +3
Mathematics · #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Algebraic structures and combinatorial models #Combinatorics (math.CO) #FOS: Mathematics
paper · pdf · doi:10.48550/arxiv.2110.15394
openalex publication_date 2021/10/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Kostka-Foulkes polynomials are Lusztig's q-analogues of weight\nmultiplicities for irreducible representations of semisimple Lie algebras. It\nhas long been known that these polynomials have non-negative coefficients. A\nstatistic on semistandard Young tableaux with partition content, called\n\charge, was used to give a combinatorial formula exhibiting this fact\nin type A. Defining a charge statistic beyond type A has been a\nlong-standing problem. Here, we take a completely new approach based on the\ndefinition of Kostka-Foulkes polynomials as an alternating sum over Kostant\npartitions, which can be thought of as formal sums of positive roots. We use a\nsign-reversing involution to obtain a positive expansion, in which the relevant\nstatistic is simply the number of parts in the Kostant partitions. The hope is\nthat the simplicity of this new crystal-like model will naturally extend to\nother classical types.\n