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Grounded partitions of type $A_1^{(1)}$ at levels 1 and 2: bijections, affine crystal graphs, and partition identities

Benedek Dombos, Jihyeug Jang

TL;DR

This work links grounded partitions to affine crystal theory for A_1^{(1)} at levels 1 and 2, establishing infinite-product generating functions at level 2 via two bijections to known partition families. It introduces a ground-state crystal model on grounded partitions and proves isomorphisms with established crystal models, enabling explicit decompositions of level-2 graphs into finite-type A_1 components. The authors derive new q-series identities by combining combinatorial bijections, major-index statistics on Yamanouchi words, and crystal-restriction techniques. These results illuminate deep connections between partition theory, crystal bases, and representation theory, offering new combinatorial proofs for Rogers–Ramanujan-type product identities and refined q-series expansions. The constructions also yield alternative models for level-2 crystals and provide a framework for future bijective proofs in related affine types.

Abstract

Grounded partitions, introduced by Dousse and Konan, are coloured partitions satisfying difference conditions given by a matrix with nonnegative integer entries. For the matrices studied in this paper, the generating functions are known to be infinite products, corresponding to the principal specialisation of characters of highest weight modules of type $A_1^{(1)}$. We give the first bijective proof that the generating functions of grounded partitions at level $2$ are infinite products. We then give a new combinatorial model for affine crystal graphs of type $A_1^{(1)}$ at level $2$, where the vertices are grounded partitions and the arrows are given by explicit bracketing rules. The grounded partition model for affine crystal graphs of highest weights $Λ_0$, $Λ_1$ and $Λ_0 + Λ_1$ gives rise to new $q$-series identities obtained by decomposing the affine crystal graphs into the crystal graphs of finite type $A_1$ via the restricted representation.

Grounded partitions of type $A_1^{(1)}$ at levels 1 and 2: bijections, affine crystal graphs, and partition identities

TL;DR

This work links grounded partitions to affine crystal theory for A_1^{(1)} at levels 1 and 2, establishing infinite-product generating functions at level 2 via two bijections to known partition families. It introduces a ground-state crystal model on grounded partitions and proves isomorphisms with established crystal models, enabling explicit decompositions of level-2 graphs into finite-type A_1 components. The authors derive new q-series identities by combining combinatorial bijections, major-index statistics on Yamanouchi words, and crystal-restriction techniques. These results illuminate deep connections between partition theory, crystal bases, and representation theory, offering new combinatorial proofs for Rogers–Ramanujan-type product identities and refined q-series expansions. The constructions also yield alternative models for level-2 crystals and provide a framework for future bijective proofs in related affine types.

Abstract

Grounded partitions, introduced by Dousse and Konan, are coloured partitions satisfying difference conditions given by a matrix with nonnegative integer entries. For the matrices studied in this paper, the generating functions are known to be infinite products, corresponding to the principal specialisation of characters of highest weight modules of type . We give the first bijective proof that the generating functions of grounded partitions at level are infinite products. We then give a new combinatorial model for affine crystal graphs of type at level , where the vertices are grounded partitions and the arrows are given by explicit bracketing rules. The grounded partition model for affine crystal graphs of highest weights , and gives rise to new -series identities obtained by decomposing the affine crystal graphs into the crystal graphs of finite type via the restricted representation.

Paper Structure

This paper contains 10 sections, 28 theorems, 89 equations, 32 figures.

Table of Contents

  1. Introduction
  2. Grounded partitions
  3. Affine crystal structures of type $A_1^{(1)}$ at levels 1 and 2
  4. Bijections at level 2
  5. Bijection between $\mathcal{P}_{2,b}$ and odd overpartitions
  6. Bijection between $\mathcal{P}_{2,a}$ and partitions without repeating even parts
  7. Affine crystal graphs on grounded partitions at level $2$
  8. Decomposition formulas at levels $1$ and $2$
  9. Major index on Yamanouchi words
  10. Decomposition formula for $\mathcal{P}_{1,b}$

Key Result

Theorem 1.2

Figures (32)

  • Figure 1: The affine crystal graph $\mathcal{B}_{1,b}^\mathcal{P}$
  • Figure 2: Affine crystal graph of highest weight $\Lambda_0 + \Lambda_1$
  • Figure 3: From grounded partition to overpartition: step 1
  • Figure 4: From grounded partition to overpartition: step 2
  • Figure 5: From grounded partition to overpartition: step 3
  • ...and 27 more figures

Theorems & Definitions (82)

  • Example 1.1
  • Theorem 1.2: Rogers--Ramanujan identities, in terms of $q$-series, RR19
  • Remark 1.3
  • Definition 1.4
  • Theorem 1.5: DHK25
  • Remark 1.6
  • Example 1.7
  • Remark 1.8
  • Remark 1.8
  • Theorem 1.9
  • ...and 72 more