Affine Jacobi-Trudi formulas and $q,t$-Rogers-Ramanujan identities

TL;DR

This paper conjectures affine and Hall-Littlewood analogues of dual Jacobi-Trudi formulas for certain Schur functions and uses these to derive $t$-analogues of classical Rogers-Ramanujan identities related to affine Lie algebra characters.

Contribution

It introduces new conjectures for affine and Hall-Littlewood analogues of dual Jacobi-Trudi formulas and derives $t$-analogues of several classical Rogers-Ramanujan identities.

Findings

Conjectured affine and Hall-Littlewood analogues of dual Jacobi-Trudi formulas.

Derived $t$-analogues of Rogers-Ramanujan and related identities.

Proved an affine dual Jacobi-Trudi formula for rectangular partitions of arbitrary height.

Abstract

We conjecture affine or Hall-Littlewood analogues of the dual Jacobi-Trudi formulas for orthogonal and symplectic Schur functions indexed by rectangular partitions of maximal height. These conjectures are then used to derive $t$-analogues of many known Rogers-Ramanujan identities for the characters of standard modules of affine Lie algebras. This includes $t$-analogues of the classical Rogers-Ramanujan identities, (some of) the Andrews-Gordon identities and the $\mathrm{C}_n^{(1)}$, $\mathrm{A}_{2n}^{(2)}$ and $\mathrm{D}_{n+2}^{(2)}$ GOW identities. We also prove an affine analogue of the dual Jacobi-Trudi formula for Schur functions indexed by rectangular partitions of arbitrary height.