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

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

2009-08-17

arxiv.org/abs/0908.2398v1

Mathematics

Representation Theory

22 pages

Scientific paper

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 $q^{(d-r)/2}(q+1)^r$.

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