Generating functions for real character degree sums of finite general linear and unitary groups

TL;DR

This paper derives generating functions for real character degree sums of finite general linear and unitary groups, providing new combinatorial proofs, identities, and explicit formulas related to Frobenius-Schur indicators.

Contribution

It introduces novel generating functions and combinatorial proofs for real character degree sums, expanding understanding of Frobenius-Schur indicators in these groups.

Findings

Proved all real-valued characters of GL(n,q) have Frobenius-Schur indicator 1.

Derived q-series identities related to character degree sums.

Expressed character degree sums for unitary groups using Hall-Littlewood functions.

Abstract

We compute generating functions for the sum of the real-valued character degrees of the finite general linear and unitary groups, through symmetric function computations. For the finite general linear group, we get a new combinatorial proof that every real-valued character has Frobenius-Schur indicator 1, and we obtain some q-series identities. For the finite unitary group, we expand the generating function in terms of values of Hall-Littlewood functions, and we obtain combinatorial expressions for the character degree sums of real-valued characters with Frobenius-Schur indicator 1 or -1.