Classical interpolation categories

TL;DR

This paper investigates tensor categories that interpolate classical group representations, comparing ultraproduct and oligomorphic group approaches, and establishes their equivalence and structural properties using finite geometry insights.

Contribution

It demonstrates the equivalence of ultraproduct and oligomorphic approaches to interpolation categories and characterizes measures on oligomorphic groups.

Findings

Ultraproduct and oligomorphic categories are shown to be equivalent.

All measures on oligomorphic groups are determined.

Structural results about ultraproduct categories are derived.

Abstract

We study tensor categories that interpolate the representation categories of finite classical groups. There are (at least) two ways to approach these categories: via ultraproducts and via oligomorphic groups. Both have strengths and weaknesses. The ultraproduct categories are easy to define, but their structure is not clear. On the other hand, the oligomorphic approach requires a certain kind of measure as an input, and the space of measures is not obvious. Furthermore, it is not a priori clear that the two approaches yield the same categories in general. We handle all of these issues: we determine all measures on the oligomorphic groups, and we show that the oligomorphic and ultraproduct categories agree, which gives us basic structural results about the latter. Our results rely upon (and in some sense repackage) enumerative results in finite geometry.