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

TL;DR

This paper develops a new fermionic formula for level-restricted generalized Kostka polynomials, linking crystal theory, rigged configurations, and coset branching functions in quantum affine algebra modules.

Contribution

It introduces an explicit characterization of level-restricted rigged configurations and derives a general fermionic formula for these polynomials, connecting to coset branching functions.

Findings

New fermionic formula for level-restricted Kostka polynomials

Explicit characterization of level-restricted rigged configurations

Computation of coset branching functions using limits of formulas

Abstract

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.