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

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 $\mathcal{Z}$ of countable dimension over afinite field equipped with a cubic form. The group of automorphisms of $\mathcal{Z}$ 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.