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Lusztig $q$-weight multiplicities and Kirillov-Reshetikhin crystals

Hyeonjae Choi, Donghyun Kim, Seung Jin Lee

TL;DR

This work resolves long-standing questions about the positivity of Lusztig $q$-weight multiplicities beyond type $A$ by providing explicit combinatorial formulas in terms of Kirillov–Reshetikhin crystals. The authors introduce new combinatorial objects (SSOT, GSSOT, SSROT) and a splitting map framework to realize Lusztig multiplicities as energy-weighted counts of crystal elements, proving type $C$ formulas for dominant weights and type $B$ formulas for spin weights. They further develop level-restricted $q$-weight multiplicities, establish their positivity, and refine the $X=K$ theorem to this setting, with detailed proofs via deformed Morris recurrences and oscilliating strip jeu de taquin. The results connect energy functions, combinatorial $R$-matrices, and LR-type tableaux, offering a robust combinatorial toolkit for weight multiplicities across nonexceptional types and opening avenues for geometric interpretations and monotonicity properties.

Abstract

Lusztig $q$-weight multiplicities extend the Kostka-Foulkes polynomials to a broader range of Lie types. In this work, we investigate these multiplicities through the framework of Kirillov-Reshetikhin crystals. Specifically, for type $C$ with dominant weights and type $B$ with dominant spin weights, we present a combinatorial formula for Lusztig $q$-weight multiplicities in terms of energy functions of Kirillov-Reshetikhin crystals, generalizing the charge statistic on semistandard Young tableaux for type $A$. Additionally, we introduce level-restricted $q$-weight multiplicities for nonexceptional types, and prove positivity by providing their combinatorial formulas.

Lusztig $q$-weight multiplicities and Kirillov-Reshetikhin crystals

TL;DR

This work resolves long-standing questions about the positivity of Lusztig -weight multiplicities beyond type by providing explicit combinatorial formulas in terms of Kirillov–Reshetikhin crystals. The authors introduce new combinatorial objects (SSOT, GSSOT, SSROT) and a splitting map framework to realize Lusztig multiplicities as energy-weighted counts of crystal elements, proving type formulas for dominant weights and type formulas for spin weights. They further develop level-restricted -weight multiplicities, establish their positivity, and refine the theorem to this setting, with detailed proofs via deformed Morris recurrences and oscilliating strip jeu de taquin. The results connect energy functions, combinatorial -matrices, and LR-type tableaux, offering a robust combinatorial toolkit for weight multiplicities across nonexceptional types and opening avenues for geometric interpretations and monotonicity properties.

Abstract

Lusztig -weight multiplicities extend the Kostka-Foulkes polynomials to a broader range of Lie types. In this work, we investigate these multiplicities through the framework of Kirillov-Reshetikhin crystals. Specifically, for type with dominant weights and type with dominant spin weights, we present a combinatorial formula for Lusztig -weight multiplicities in terms of energy functions of Kirillov-Reshetikhin crystals, generalizing the charge statistic on semistandard Young tableaux for type . Additionally, we introduce level-restricted -weight multiplicities for nonexceptional types, and prove positivity by providing their combinatorial formulas.
Paper Structure (30 sections, 56 theorems, 216 equations, 1 table)

This paper contains 30 sections, 56 theorems, 216 equations, 1 table.

Table of Contents

  1. Introduction
  2. Organization
  3. Preliminaries
  4. Partitions and orthogonal complements
  5. Crystals
  6. KR crystals
  7. Splitting map
  8. Combinatorial objects and embedding into KR crystals
  9. Lusztig $q$-weight multiplicities
  10. Deforming Morris type recurrence formula
  11. Oscillating horizontal strip jeu de taquin
  12. Proof of Theorem \ref{['thm: B lusztig']}
  13. level-restricted $q$-weight multiplicities
  14. Refining $X=K$ theorem for $B_{\mu}(\diamond)$
  15. $\diamond=\hbox{\ydiagram{1}}$ or $\hbox{\ydiagram{2}}$
  16. ...and 15 more sections

Key Result

Lemma 2.1

LOS2012 For $b \in B_\mu$, we have $\overline{D}\left( S_\mu(b) \right)=\overline{D}(b)$.

Theorems & Definitions (140)

  • Lemma 2.1
  • Remark 2.2
  • Remark 2.3
  • Lemma 2.4
  • Definition 3.1
  • Definition 3.2
  • Definition 3.3
  • Example 3.4
  • Lemma 3.5
  • proof
  • ...and 130 more