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A new grounded partition identity of type $D_4^{(3)}$

Benedek Dombos

TL;DR

The paper establishes a new Rogers–Ramanujan–type identity arising from the exceptional twisted affine algebra $D_4^{(3)}$ by equating a product side, obtained via Lepowsky's principal-specialisation formula, with a sum side derived from grounded partitions through a $D_4^{(3)}$ perfect crystal. The product side yields a generating function for partitions with parts congruent to $1$ or $5$ modulo $6$, while the sum side is realized as a grounded-partition generating function using energy matrices and the KMN2 character formula. The work further develops recursions to analyze refinements by colour tracking, proving that maintaining colours prevents an infinite product unless all colours are specialised, thereby highlighting fundamental limits of color-refinement in this framework. Overall, this is the first Rogers–Ramanujan-type identity connected to an exceptional affine type via perfect crystals and grounded partitions, deepening the link between representation theory and partition theory.

Abstract

In this paper, we prove a new Rogers--Ramanujan-type identity, involving grounded partitions, by computing a character of the affine Kac--Moody algebra $D_4^{(3)}$ in two different ways. The product side is derived using Lepowsky's product formula, while the sum side is obtained using perfect crystals with a technique of Dousse and Konan.

A new grounded partition identity of type $D_4^{(3)}$

TL;DR

The paper establishes a new Rogers–Ramanujan–type identity arising from the exceptional twisted affine algebra by equating a product side, obtained via Lepowsky's principal-specialisation formula, with a sum side derived from grounded partitions through a perfect crystal. The product side yields a generating function for partitions with parts congruent to or modulo , while the sum side is realized as a grounded-partition generating function using energy matrices and the KMN2 character formula. The work further develops recursions to analyze refinements by colour tracking, proving that maintaining colours prevents an infinite product unless all colours are specialised, thereby highlighting fundamental limits of color-refinement in this framework. Overall, this is the first Rogers–Ramanujan-type identity connected to an exceptional affine type via perfect crystals and grounded partitions, deepening the link between representation theory and partition theory.

Abstract

In this paper, we prove a new Rogers--Ramanujan-type identity, involving grounded partitions, by computing a character of the affine Kac--Moody algebra in two different ways. The product side is derived using Lepowsky's product formula, while the sum side is obtained using perfect crystals with a technique of Dousse and Konan.

Paper Structure

This paper contains 8 sections, 8 theorems, 54 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Lepowsky's product formula
  4. Lepowsky's product formula
  5. Perfect crystals
  6. Congruence conditions
  7. Difference conditions
  8. Recursions

Key Result

Theorem 1.1

Figures (7)

  • Figure 1: Generalised Cartan matrices of $G_2^{(1)}$ and $D_4^{(3)}$
  • Figure 2: $G_2$ root system
  • Figure 3: Perfect crystal $\mathcal{B}$ of level 1 of type $D_4^{(3)}$
  • Figure 4: weights and levels of the vertices of $\mathcal{B}$
  • Figure 5: the crystal $\mathcal{B} \otimes \mathcal{B}$
  • ...and 2 more figures

Theorems & Definitions (16)

  • Theorem 1.1: Rogers--Ramanujan identities, in terms of $q$-series, RR19
  • Definition 1.2
  • Remark 1.3
  • Theorem 1.4: DHK25
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 2.1: Weyl--Kac character formula Kac90 Theorem 10.4
  • Theorem 2.2: Lepowsky's product formula, Lep78
  • Remark 2.3
  • Definition 2.4
  • ...and 6 more