Totally orthogonal finite simple groups

2018/11/13 by C. Ryan Vinroot, Vinroot, C. Ryan
Computer Science · Mathematics · #05A15 #20C33 #20D05 #Advanced Algebra and Geometry #Coding theory and cryptography #FOS: Mathematics #Finite Group Theory Research #Group Theory (math.GR) #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.1811.05343

openalex publication_date 2018/11/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We prove that if G is a finite simple group, then all irreducible complex representations of G by be realized over the real numbers if and only if every element of G may be written as a product of two involutions in G. This follows from our result that if q is a power of 2, then all irreducible complex representations of the orthogonal groups O^±(2n, \mathbbF_q) may be realized over the real numbers. We also obtain generating functions for the sums of degrees of several sets of unipotent characters of finite orthogonal groups, and we obtain a twisted version of our main result for a broad family of finite classical groups.

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