Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 22 (2026), 062, 50 pages      arXiv:2511.17034      https://doi.org/10.3842/SIGMA.2026.062
Contribution to the Special Issue on Recent Advances in Vertex Operator Algebras in honor of James Lepowsky

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

S. Ole Warnaar
School of Mathematics and Physics, The University of Queensland, Brisbane, Australia

Received November 24, 2025, in final form June 05, 2026; Published online June 25, 2026

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.

Key words: affine root systems; character formulas for standard modules; cylindric Schur functions; Hall-Littlewood polynomials; Jacobi-Trudi identities; Rogers-Ramanujan identities; theta function identities.

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