Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

S. Ole Warnaar

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.

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

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 -analogues of many known Rogers-Ramanujan identities for the characters of standard modules of affine Lie algebras. This includes -analogues of the classical Rogers-Ramanujan identities, (some of) the Andrews-Gordon identities and the , and GOW identities. We also prove an affine analogue of the dual Jacobi-Trudi formula for Schur functions indexed by rectangular partitions of arbitrary height.

Paper Structure

This paper contains 17 sections, 40 theorems, 130 equations, 1 figure.

Table of Contents

  1. Dedicatory
  2. Introduction
  3. The Jacobi--Trudi and dual Jacobi--Trudi formulas
  4. Main results and conjectures
  5. Outline
  6. Symmetric functions
  7. (Cylindric) Schur functions
  8. Hall--Littlewood polynomials for $\mathrm{GL}(n,\mathbb{C})$
  9. The $\mathrm{B}_n$, $\mathrm{C}_n$ and $\mathrm{BC}_n$ Hall--Littlewood polynomials
  10. Determinant identities for elementary symmetric functions
  11. Proof, special cases and generalisations of Theorem \ref{['Thm_GLn']}
  12. Proofs of special cases of Conjectures \ref{['Con_JTB']}--\ref{['Con_JTBC']}
  13. $q,t$-Rogers--Ramanujan identities
  14. From Jacobi--Trudi to $q,t$-Rogers--Ramanujan identities
  15. From $q,t$-Rogers--Ramanujan to Rogers--Ramanujan identities for standard modules
  16. ...and 2 more sections

Key Result

Theorem 1.1

Let $k$ be a positive integer, $r$ a nonnegative integer and $x=(x_1,\dots,x_n)$. Then

Figures (1)

  • Figure 1: The Dynkin diagrams of the affine Lie algebras $\mathrm{A}_{2n}^{(2)}$ ($n\geqslant 1$), $\mathrm{B}_n^{(1)}$ ($n\geqslant 3$), $\mathrm{A}_{2n-1}^{(2)}$ ($n\geqslant 3$), $\mathrm{C}_n^{(1)}$ ($n\geqslant 2$), $\mathrm{D}_{n+1}^{(2)}$ ($n\geqslant 2$) and $\mathrm{D}_n^{(1)}$ ($n\geqslant 4$).

Theorems & Definitions (43)

  • Theorem 1.1
  • Conjecture 1.2
  • Conjecture 1.3
  • Conjecture 1.4
  • Theorem 1.5: $q,t$-Rogers--Ramanujan-type identity for $\mathrm{C}_k^{(1)}$
  • Theorem 1.6: $q,t$-Rogers--Ramanujan-type identities for $\mathrm{A}_{2k}^{(2)}$
  • Theorem 1.7: $q,t$-Rogers--Ramanujan identity for $\mathrm{D}_{k+1}^{(2)}$
  • Theorem 1.8: $q,t$-Rogers--Ramanujan identity for $\mathrm{A}_{2k-1}^{(2)}$
  • Proposition 2.1
  • Theorem 2.2: Kirillov Kirillov00
  • ...and 33 more