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

TL;DR

This paper explores the structure of oligomorphic groups through categories of partial bijections, exemplified by automorphism groups of ultrahomogeneous cubic spaces over finite fields, and relates them to unitary representation classification.

Contribution

It introduces a framework for multiplication of double cosets in certain oligomorphic groups and characterizes categories of partial bijections, with a detailed example involving cubic spaces.

Findings

Established a version of double coset multiplication for specific oligomorphic groups.

Realized categories of double cosets as categories of partial bijections.

Classified unitary representations of automorphism groups of ultrahomogeneous cubic spaces.

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.