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An $\mathrm{A}_2$ Bailey tree and $\mathrm{A}_2^{(1)}$ Rogers-Ramanujan-type identities

S. Ole Warnaar

TL;DR

This work develops a four-parameter $ ext{A}_2$ Bailey tree that extends the $ ext{A}_2$ Bailey chain, enabling a case-free proof of the Kanade–Russell conjecture for a broad three-parameter family of $ ext{A}_2^{(1)}$ Rogers–Ramanujan-type identities. It also yields an $ ext{A}_2^{(1)}$ analogue of the Andrews–Gordon identities and derives Rogers–Selberg-type character identities for principal subspaces of $ ext{A}_2^{(1)}$, including a level-$k$ dominant-integral weights generalization. The paper further connects these multisum identities to the characters of the $ ext{W}_3(3,K)$ vertex operator algebra and to principal-subspace character formulas, and concludes with an outlook on extending the framework to higher rank, $ ext{A}_{r-1}^{(1)}$, and related Kostka-polynomial structures. The results illuminate deep links between high-rank Bailey theory, affine Lie algebras, and combinatorial partition theory in the Rogers–Ramanujan landscape.

Abstract

The $\mathrm{A}_2$ Bailey chain of Andrews, Schilling and the author is extended to a four-parameter $\mathrm{A}_2$ Bailey tree. As main application of this tree, we prove the Kanade-Russell conjecture for a three-parameter family of Rogers-Ramanujan-type identities related to the principal characters of the affine Lie algebra $\mathrm{A}_2^{(1)}$. Combined with known $q$-series results, this further implies an $\mathrm{A}_2^{(1)}$-analogue of the celebrated Andrews-Gordon $q$-series identities. We also use the $\mathrm{A}_2$ Bailey tree to prove a Rogers-Selberg-type identity for the characters of the principal subspaces of $\mathrm{A}_2^{(1)}$ indexed by arbitrary level-$k$ dominant integral weights $λ$. This generalises a result of Feigin, Feigin, Jimbo, Miwa and Mukhin for $λ=kΛ_0$.

An $\mathrm{A}_2$ Bailey tree and $\mathrm{A}_2^{(1)}$ Rogers-Ramanujan-type identities

TL;DR

This work develops a four-parameter Bailey tree that extends the Bailey chain, enabling a case-free proof of the Kanade–Russell conjecture for a broad three-parameter family of Rogers–Ramanujan-type identities. It also yields an analogue of the Andrews–Gordon identities and derives Rogers–Selberg-type character identities for principal subspaces of , including a level- dominant-integral weights generalization. The paper further connects these multisum identities to the characters of the vertex operator algebra and to principal-subspace character formulas, and concludes with an outlook on extending the framework to higher rank, , and related Kostka-polynomial structures. The results illuminate deep links between high-rank Bailey theory, affine Lie algebras, and combinatorial partition theory in the Rogers–Ramanujan landscape.

Abstract

The Bailey chain of Andrews, Schilling and the author is extended to a four-parameter Bailey tree. As main application of this tree, we prove the Kanade-Russell conjecture for a three-parameter family of Rogers-Ramanujan-type identities related to the principal characters of the affine Lie algebra . Combined with known -series results, this further implies an -analogue of the celebrated Andrews-Gordon -series identities. We also use the Bailey tree to prove a Rogers-Selberg-type identity for the characters of the principal subspaces of indexed by arbitrary level- dominant integral weights . This generalises a result of Feigin, Feigin, Jimbo, Miwa and Mukhin for .
Paper Structure (12 sections, 26 theorems, 237 equations)

This paper contains 12 sections, 26 theorems, 237 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The $\mathrm{A}_1$ Bailey lemma
  4. The $\mathrm{A}_2$ Bailey tree
  5. A Bailey tree for $\mathrm{A}_2$
  6. $\mathrm{A}_2$ Bailey pairs
  7. Proof of the Kanade--Russell conjecture
  8. Below-the-line identities
  9. Character identities for the $\mathcal{W}_3(3,K)$ vertex operator algebra
  10. Character formulas for principal subspaces of $\mathrm{A}_2^{(1)}$
  11. Outlook
  12. New proof of \ref{['Eq_page692']}

Key Result

Theorem 1.2

The Kanade--Russell conjecture holds for all moduli.

Theorems & Definitions (42)

  • Conjecture 1.1: Kanade--Russell
  • Theorem 1.2
  • Theorem 1.3: $\mathrm{A}_2^{(1)}$ Andrews--Gordon identities; $b=0$ case
  • Theorem 1.4
  • Corollary 1.5
  • Lemma 3.1: $\mathrm{A}_1$ Bailey chain
  • proof
  • Corollary 3.2
  • proof
  • Lemma 3.3: $\mathrm{A}_1$ Bailey tree
  • ...and 32 more