Affine Jacobi-Trudi Identities and $q,t$-Rogers-Ramanujan Identities

Abstract

We conjecture affine or Hall-Littlewood analogues of the dual Jacobi-Trudi identities 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 identity for Schur functions indexed by rectangular partitions of arbitrary height.