Irreducible characters of the finite group GLn(q) were determined by Green in a remarkable paper that has influenced representation theory greatly. In this talk, I will discuss a vertex algebraic approach to construct and compute all complex irreducible characters of GLn(q). Green’s theory is recovered and enhanced under the realization of the Grothendieck ring of representations R(G)=⨁n≥0R(GLn(q)) as two isomorphic Fock spaces. Under this picture, 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. This offers a simplification to identify the Fock space R(G) as the Hall algebra of symmetric functions. We will also discuss how to compute the characters in general.
This is joint work with Y. Wu.
This video was part of the Southeastern Lie Theory Workshop XIV on quantum structures in Lie theory.
