Fermionic formula for 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_{𝛌, 𝛍}(𝑡), indexed by two double partitions 𝛌, 𝛍, are polynomials in t introduced as a generalization of Kostka polynomials. In the present paper, we consider 𝐾_{𝛌, 𝛍}(𝑡) in the special case where 𝛍 = (−, 𝛍″). We formulate a 1𝐷 sum and a fermionic formula for 𝐾_{𝛌, 𝛍}(𝑡), as a generalization of the case of ordinary Kostka polynomials. Then we prove an analogue of the 𝑋 = 𝑀 conjecture.