Oligomorphic groups, categories of partial bijections, and ultrahomogeneous cubic spaces over finite fields

2023/08/25 by Neretin, Yury A.
#18B10 #22A25 #22F50 #Category Theory (math.CT) #FOS: Mathematics #Group Theory (math.GR) #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.2308.13247

Abstract

We show that for certain class of oligomorphic groups there is a version of multiplication of double cosets in the Ismagilov--Olshanski sense. Categories of (reduced) double cosets are realized as certain categories of partial bijections. As an example, we consider the ultrahomogeneous linear space Z of countable dimension over afinite field equipped with a cubic form. The group of automorphisms of Z is an oligomorphic group, we describe its open subgroups. According Tsankov, this gives a classification of its unitary representations. The category of reduced double cosets in this case is the category of partial linear bijections of finite-dimensional cubic spaces preserving cubic forms.

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