The semi-linear representation theory of the infinite symmetric group

TL;DR

This paper investigates the structure of smooth semilinear representations of the infinite symmetric group over a field of rational functions, revealing their classification, properties, and an equivalence to a simpler algebraic category.

Contribution

It introduces a classification of injective objects, computes the Grothendieck group, and establishes an equivalence to a simpler linear algebraic category.

Findings

Classification of injective objects in the category

Finiteness of the injective dimension

Equivalence to a simpler linear algebraic category

Abstract

We study the category $\mathcal{A}$ of smooth semilinear representations of the infinite symmetric group over the field of rational functions in infinitely many variables. We establish a number of results about the structure of $\mathcal{A}$, e.g., classification of injective objects, finiteness of injective dimension, computation of the Grothendieck group, and so on. We also prove that $\mathcal{A}$ is (essentially) equivalent to a simpler linear algebraic category $\mathcal{B}$, which makes many properties of $\mathcal{A}$ transparent.