On semilinear representations of the infinite symmetric group

TL;DR

This paper investigates smooth semilinear representations of infinite permutation groups, revealing that under certain conditions, irreducible representations are essentially one-dimensional, extending classical results to a broader context.

Contribution

It extends the understanding of semilinear representations of infinite symmetric groups, suggesting an analogue of Hilbert's Theorem 90 for these groups.

Findings

Irreducible smooth semilinear representations are one-dimensional under faithful action.

Many results are known for linear representations, now extended to semilinear case.

The work indicates a potential generalization of classical theorems to infinite permutation groups.

Abstract

In this note the smooth (i.e. with open stabilizers) linear and {\sl semilinear} representations of certain permutation groups (such as infinite symmetric group or automorphism group of an infinite-dimensional vector space over a finite field) are studied. Many results here are well-known to the experts, at least in the case of {\sl linear representations} of symmetric group. The presented results suggest, in particular, that an analogue of Hilbert's Theorem 90 should hold: in the case of faithful action of the group on the base field the irreducible smooth semilinear representations are one-dimensional (and trivial in appropriate sense).