New fermionic formula for unrestricted Kostka polynomials

Abstract

A new fermionic formula for the unrestricted Kostka polynomials of type $A_{n-1}^{(1)}$ is presented. This formula is different from the one given by Hatayama et al. and is valid for all crystal paths based on Kirillov-Reshetihkin modules, not just for the symmetric and anti-symmetric 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.Comment: 35 pages; reference adde

Kirillov-Schilling-Shimozono Bijection as Energy Functions of Crystals

“…Let W (r) s be a U q (g) Kirillov-Reshetikhin module, where we shall consider the case g = A (1) n . The module W (r) s is indexed by a Dynkin node r ∈ {1, 2, .…”

“…Recently, the original KSS bijection was extended to include non-highest weight elements [23,1]. The combinatorial procedures for φ and φ −1 are the formal extension of the original ones.…”

“…Then our main result (Theorem 3.3) states that by taking differences of these E l or e i 's, we can derive alternative algorithm of the map φ. This is a generalization of the procedure given in [20] where the case A (1) 1 is treated.…”

Making Research on Symmetric Functions with MuPAD-Combinat

“…In order to generalize Kostka polynomials, there exist other combinatorial ways (with unrestricted rigged configurations recently introduced by L. Deka and A. Schilling in [1]) and algebraic ways (using the theory of crystal bases for quantum groups of type A n ) to generalize Kostka polynomials. Other generalizations are given by M. Zabrocki using creation operators in [16] and by L. Lapointe and J. Morse introducing a t-deformation of k-Schur functions in [9].…”

“…combinat::skewRiggings L. Deka and A. Schilling introduced in [1] a new kind of rigged configurations, namely the unrestricted ones. They were implemented as an independent C++ program (file: FromOneCrystalPath.cc, headers: FromOneCrystalPath.h).…”