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

2025/11/21 by Warnaar, S. Ole
#05E05 #05E10 #11P84 #17B67 #33D52 #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT) #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.2511.17034

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 C_n⁽¹⁾, A_2n⁽²⁾ and 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.

Citations

• Tight cylindric partitions
• Remarks on the conjectures of Capparelli, Meurman, Primc and Primc
• Companions to the Andrews-Gordon and Andrews-Bressoud identities, and recent conjectures of Capparelli, Meurman, Primc, and Primc
• Bounded Littlewood identities for cylindric Schur functions
• New partition identities for odd w odd
• Characters of level 1 standard modules of C_n⁽¹⁾ as generating functions for generalised partitions
• Positivity of Cylindric skew Schur functions
• IdentityFinder and some new identities of Rogers-Ramanujan type
• Affine approach to quantum Schubert calculus
• Change of base in Bailey pairs

Related

• The A_2n⁽²⁾ Rogers-Ramanujan identities
• Remarks on the conjectures of Capparelli, Meurman, Primc and Primc
• Revisiting Jacobi-Trudi identities via the BGG category O
• Staircases to analytic sum-sides for many new integer partition identities of Rogers-Ramanujan type
• The Rogers-Ramanujan Identities, the Finite General Linear Groups, and the Hall-Littlewood Polynomials