Character degree sums and real representations of finite classical groups of odd characteristic

TL;DR

This paper investigates real-valued characters of finite classical groups over finite fields of odd characteristic, establishing their correspondence with real representations, calculating their sum of dimensions, and providing bounds on sums of irreducible character degrees.

Contribution

It proves that all real-valued irreducible characters correspond to real representations and derives bounds on sums of character degrees for classical groups over finite fields.

Findings

Real-valued irreducible characters are associated with real representations.

Sum of dimensions of real representations is explicitly calculated.

Upper and lower bounds for sums of irreducible character degrees are established.

Abstract

Let $\mathbb{F}_q$ be a finite field with $q$ elements, where $q$ is the power of an odd prime, and let $\mathrm{GSp}(2n, \mathbb{F}_q)$ and $\mathrm{GO}^{\pm}(2n, \mathbb{F}_q)$ denote the symplectic and orthogonal groups of similitudes over $\mathbb{F}_q$, respectively. We prove that every real-valued irreducible character of $\mathrm{GSp}(2n, \mathbb{F}_q)$ or $\mathrm{GO}^{\pm}(2n, \mathbb{F}_q)$ is the character of a real representation, and we find the sum of the dimensions of the real representations of each of these groups. We also show that if $\boldsymbol{G}$ is a classical connected group defined over $\mathbb{F}_q$ with connected center, with dimension $d$ and rank $r$, then the sum of the degrees of the irreducible characters of $\boldsymbol{G}(\mathbb{F}_q)$ is bounded above by $(q+1)^{(d+r)/2}$. Finally, we show that if $\boldsymbol{G}$ is any connected reductive group defined over $\mathbb{F}_q$, for any $q$, the sum of the degrees of the irreducible characters of $\boldsymbol{G}(\FF_q)$ is bounded below by $q^{(d-r)/2}(q-1)^r$. We conjecture that this sum can always be bounded above by $q^{(d-r)/2}(q+1)^r$.