Partition identities from higher level crystals of $A_1^{(1)}$

TL;DR

This work addresses partition identities derived from higher-level crystals of $A_1^{(1)}$ by marrying perfect crystals, energy matrices, and grounded/multi-grounded partitions to produce sign-free character formulas and a new family of companion partition identities to the Meurman–Primc identities. The core contributions include (i) a pair of partition identities with absolute-value difference conditions whose generating functions match the Andrews–Gordon/Bressoud product sides, (ii) manifestly positive non-specialised character formulas for all level‑2 standard modules of $A_1^{(1)}$, and (iii) explicit crystal–combinatorial correspondences via ground-state paths and bijections between grounded partitions and crystal paths. The results showcase a robust framework for converting crystal data into combinatorial partition representations and connect to the Weyl–Kac principal specialization. Overall, the paper advances a bridge between crystal bases, vertex operator algebras, and partition theory, with potential extensions to $A_n^{(1)}$ at arbitrary levels.

Abstract

We study perfect crystals for the standard modules of the affine Lie algebra $A_1^{(1)}$ at all levels using the theory of multi-grounded partitions. We prove a family of partition identities which are reminiscent of the Andrews-Gordon identities and companions to the Meurman-Primc identities, but with simple difference conditions involving absolute values.

Figures (2)

• Figure 1: The level $n$ perfect crystal $\mathcal{B}_n$ of $A_1^{(1)}$
• Figure 2: The level $2$ perfect crystal of $A_1^{(1)}$

Theorems & Definitions (20)

• Theorem 1.1: The Rogers--Ramanujan identities
• Theorem 1.2: Rogers--Ramanujan identities, combinatorial version
• Theorem 1.3: Andrews--Gordon identities
• Theorem 1.4: Bressoud
• Theorem 1.5: Meurman--Primc 1999
• Theorem 1.6
• Proposition 1.7
• Remark 1.8
• Theorem 2.1: (KMN)$^2$ crystal base character formula KMN2a
• Definition 2.2