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

2009/08/17 by Vinroot, C. Ryan
#20C33 #20G40 #FOS: Mathematics #Group Theory (math.GR) #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.0908.2398

Abstract

Let \mathbbF_q be a finite field with q elements, where q is the power of an odd prime, and let GSp(2n, \mathbbF_q) and GO^±(2n, \mathbbF_q) denote the symplectic and orthogonal groups of similitudes over \mathbbF_q, respectively. We prove that every real-valued irreducible character of GSp(2n, \mathbbF_q) or GO^±(2n, \mathbbF_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 \boldsymbolG is a classical connected group defined over \mathbbF_q with connected center, with dimension d and rank r, then the sum of the degrees of the irreducible characters of \boldsymbolG(\mathbbF_q) is bounded above by (q+1)^(d+r)/2. Finally, we show that if \boldsymbolG is any connected reductive group defined over \mathbbF_q, for any q, the sum of the degrees of the irreducible characters of \boldsymbolG(\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.

Related

• Real representations of finite symplectic groups over fields of\n characteristic two
• Totally orthogonal finite simple groups
• A uniform upper bound for the character degree sums and Gelfand-Graev-like characters for finite simple groups
• Values of characters sums for finite unitary groups
• Generating functions for real character degree sums of finite general linear and unitary groups