Fermionic formula for double Kostka polynomials

TL;DR

This paper extends the fermionic formula and the $X=M$ conjecture to double Kostka polynomials, providing new combinatorial formulas and proving their equivalence in a special case.

Contribution

It introduces a fermionic formula and a $1D$ sum for double Kostka polynomials in a specific case, generalizing known results for ordinary Kostka polynomials.

Findings

Formulated a $1D$ sum and fermionic formula for double Kostka polynomials.

Proved an analogue of the $X=M$ conjecture for these polynomials.

Extended combinatorial and algebraic understanding of double Kostka polynomials.

Abstract

The $X=M$ conjecture asserts that the $1D$ sum and the fermionic formula coincide up to some constant power. In the case of type $A,$ both the $1D$ sum and the fermionic formula are closely related to Kostka polynomials. Double Kostka polynomials $K_{\Bla,\Bmu}(t),$ indexed by two double partitions $\Bla,\Bmu,$ are polynomials in $t$ introduced as a generalization of Kostka polynomials. In the present paper, we consider $K_{\Bla,\Bmu}(t)$ in the special case where $\Bmu=(-,\mu'').$ We formulate a $1D$ sum and a fermionic formula for $K_{\Bla,\Bmu}(t),$ as a generalization of the case of ordinary Kostka polynomials. Then we prove an analogue of the $X=M$ conjecture.