Characters of \(\mathrm{GL}_n (\mathbb{F}_q)\) and vertex operators (Q6558481)

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.