Characters of \(\mathrm{GL}_n (\mathbb{F}_q)\) and vertex operators
type: Item
instance of: scholarly article
title: Characters of \(\mathrm{GL}_n (\mathbb{F}_q)\) and vertex operators
zbMATH Open document ID: 1542.20056
author: Naihuan Jing
author: Yu Wu
published in: Journal of Algebra
publication date: 2024-06-19T00:00:00Z
review text: Let \(\mathrm{GL}_{n}(\mathbb{F}_{q})\) be the general linear group of \(n\times n\) invertible matrices over the finite field \(\mathbb{F}_{q}\). In a seminal paper, \textit{J. A. Green} [Trans. Am. Math. Soc. 80, 402-447 (1955; Zbl 0068.25605)] determined all irreducible characters of \(\mathrm{GL}_{n}(\mathbb{F}_{q})\) by \(q\) parabolic induction, generalizing Frobenius' theory of the symmetric groups.\N\NIn the paper under review, the authors present a vertex operator approach to construct and compute all complex irreducible characters of \(\mathrm{GL}_{n}(\mathbb{F}_{q})\). Green's theory is recovered and enhanced under the realization of the Grothendieck ring of representations \(R_{G}=\bigoplus_{n \geq 0} R(\mathrm{GL}_{n}(\mathbb{F}_{q}))\) as two isomorphic Fock spaces associated to two infinite-dimensional \(F\)-equivariant Heisenberg Lie algebras \(\widehat{\mathfrak{h}}_{\widehat{\overline{\mathbb{F}}}_{q}}\) and \(\widehat{\mathfrak{h}}_{\overline{\mathbb{F}}_{q}}\), where \(F\) is the Frobenius automorphism of the algebraically closed field \(\overline{\mathbb{F}}_{q}\). The irreducible characters are realized by the Bernstein vertex operators for Schur functions, the characteristic functions of the conjugacy classes are realized by the vertex operators for the Hall-Littlewood functions and the character table is completely given by matrix coefficients of vertex operators of these two types. One of the features of the authors' approach is a simpler identification of the Fock space \(R_{G}\) as the Hall algebra of symmetric functions via vertex operator calculus and another is that they are able to compute in general the character table.
reviewed by: Enrico Jabara
Mathematics Subject Classification ID: 20C15
Mathematics Subject Classification ID: 20C33
Mathematics Subject Classification ID: 17B69
Mathematics Subject Classification ID: 05E10
zbMATH DE Number: 7868223
zbMATH Keywords: finite general linear groups
zbMATH Keywords: characters
zbMATH Keywords: vertex operators
zbMATH Keywords: Hall-Littlewood functions
zbMATH Keywords: Schur functions
MaRDI profile type: Publication
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DOI: 10.1016/J.JALGEBRA.2024.04.020
description: scientific article; zbMATH DE number 7868223