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On the computation of the canonical basis for irreducible highest weight $U_q (\mathfrak{gl}_{\infty})$-module

Nicolas Jacon, Abel Lacabanne

TL;DR

This work advances the computation of canonical bases in higher-level Fock spaces for $U_q(\mathfrak{gl}_{\infty})$ by extending the Leclerc--Miyachi framework to arbitrary levels using symbol combinatorics. It introduces a column-removal principle and asymptotic factorization results that reduce complex basis elements to simpler blocks, and proves monomiality in several new regimes, including the $l=3$ case and various ordered/structured symbols. The developed techniques yield explicit, closed formulas for canonical basis elements and have direct implications for Calogero--Moser cellular characters and Ariki--Koike algebra decomposition matrices, linking representation theory to combinatorial and cellular structures. Overall, the paper broadens the practical computability of canonical bases in infinite-rank quantum groups and enriches the interplay between combinatorics, Calogero--Moser theory, and Hecke-algebra-type phenomena.

Abstract

We study canonical basis elements in higher-level Fock spaces associated with the quantum group $U_q(\mathfrak{gl}_\infty)$, which are conjecturally related to Calogero-Moser theory for complex reflection groups. We generalize the Leclerc-Miyachi formula to arbitrary levels by introducing new explicit constructions based on symbols, including a column removal theorem and closed formulas in several cases. These results provide explicit descriptions of canonical basis elements with applications to Calogero-Moser cellular characters and to the decomposition matrices of Ariki-Koike algebras.

On the computation of the canonical basis for irreducible highest weight $U_q (\mathfrak{gl}_{\infty})$-module

TL;DR

This work advances the computation of canonical bases in higher-level Fock spaces for by extending the Leclerc--Miyachi framework to arbitrary levels using symbol combinatorics. It introduces a column-removal principle and asymptotic factorization results that reduce complex basis elements to simpler blocks, and proves monomiality in several new regimes, including the case and various ordered/structured symbols. The developed techniques yield explicit, closed formulas for canonical basis elements and have direct implications for Calogero--Moser cellular characters and Ariki--Koike algebra decomposition matrices, linking representation theory to combinatorial and cellular structures. Overall, the paper broadens the practical computability of canonical bases in infinite-rank quantum groups and enriches the interplay between combinatorics, Calogero--Moser theory, and Hecke-algebra-type phenomena.

Abstract

We study canonical basis elements in higher-level Fock spaces associated with the quantum group , which are conjecturally related to Calogero-Moser theory for complex reflection groups. We generalize the Leclerc-Miyachi formula to arbitrary levels by introducing new explicit constructions based on symbols, including a column removal theorem and closed formulas in several cases. These results provide explicit descriptions of canonical basis elements with applications to Calogero-Moser cellular characters and to the decomposition matrices of Ariki-Koike algebras.
Paper Structure (13 sections, 16 theorems, 128 equations)

This paper contains 13 sections, 16 theorems, 128 equations.

Table of Contents

  1. Introduction
  2. The algebra $U_q(\mathfrak{gl}_{\infty})$ and the Fock space $\mathcal{F}_{\mathbf{v}}$
  3. Definitions
  4. Multipartitions and symbols
  5. Action on the Fock space
  6. Monomial and canonical bases
  7. Removal Theorem
  8. Asymptotic cases
  9. Column Removal Theorem for symbols
  10. Computation of canonical basis elements
  11. The case $l=3$
  12. Ordered symbol
  13. A generalization of Leclerc--Miyachi's formula

Key Result

Proposition 2.4

Let $S=(\beta^1,\ldots,\beta^l)$ be a symbol. Let $i\in \mathbb{Z}$ and let $a\in \mathbb{N}$. Then we have: where the sum is taken over all the symbol $S'=(\gamma^1,\ldots,\gamma^l)$ obtained from $S$ by replacing exactly $a$ entries $i+1$ with $i$, say in rows $j_1, \ldots, j_a$, and where Similarly, we have: where the sum is taken over all the symbol $S'=(\gamma^1,\ldots,\gamma^l)$ obtained

Theorems & Definitions (47)

  • Example 2.1
  • Example 2.2
  • Definition 2.3
  • Proposition 2.4
  • Proposition 2.5
  • Example 2.6
  • Theorem 2.7
  • Remark 2.8
  • Theorem 2.9: HL
  • Example 2.10
  • ...and 37 more