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Characters of Ariki--Koike algebras

Deke Zhao

Abstract

In this paper, we prove the Regev formulae for the characters of the Ariki--Koike algebras by applying the Schur--Sergeev reciprocity between the quantum superalgebras and the Ariki--Koike algebras, which is a generalization of the formulas in (D. Zhao, Israel J. Math. 229 (2019): 67--83 ). As a corollary, we provide the Regev formulae for the characters of the complex reflection group of type $G(m,1,n)$, which is a generalization of the formulas in (A. Regev, Israel J. Math. 195 (2013): 31--35).

Characters of Ariki--Koike algebras

Abstract

In this paper, we prove the Regev formulae for the characters of the Ariki--Koike algebras by applying the Schur--Sergeev reciprocity between the quantum superalgebras and the Ariki--Koike algebras, which is a generalization of the formulas in (D. Zhao, Israel J. Math. 229 (2019): 67--83 ). As a corollary, we provide the Regev formulae for the characters of the complex reflection group of type , which is a generalization of the formulas in (A. Regev, Israel J. Math. 195 (2013): 31--35).

Paper Structure

This paper contains 6 sections, 10 theorems, 67 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Standard elements
  4. Traces of permutation super representations
  5. $(\boldsymbol{k},\boldsymbol{\ell})$-hook multipartitions
  6. Specializations

Key Result

Theorem 1.1

Let $\boldsymbol{\mu}=(\mu^{(1)};\ldots;\mu^{(m)})$ be an $m$-multipartition of $n$. Then where $\tilde{\ell}(\boldsymbol{\alpha};\boldsymbol{\beta})=\max\{i|\ell(\alpha^{(i)},\beta^{(i)})>0\}$.

Theorems & Definitions (20)

  • Theorem 1.1
  • Corollary 1.2
  • Corollary 1.3
  • Corollary 1.4
  • Theorem 1.5
  • Theorem 2.9: Z24
  • Remark 2.10
  • Remark 3.6
  • Lemma 4.1
  • proof
  • ...and 10 more