Semilinear representations of symmetric groups and of automorphism groups of universal domains

Abstract

Let K be a field and G be a group of its automorphisms endowed with the compact-open topology, cf. Sect. 1.1. If G is precompact then K is a generator of the category of smooth (i.e. with open stabilizers) K-semilinear representations of G, cf. Proposition 1.1. There are non-semisimple smooth semilinear representations of G over K if G is not precompact. In this note the smooth semilinear representations of the group $\mathop {\mathfrak {S}}\nolimits _{\Psi }$ of all permutations of an infinite set $\Psi$ are studied. Let k be a field and $k(\Psi )$ be the field freely generated over k by the set $\Psi$ (endowed with the natural $\mathop {\mathfrak {S}}\nolimits _{\Psi }$-action). One of principal results describes the Gabriel spectrum of the category of smooth $k(\Psi )$-semilinear representations of $\mathop {\mathfrak {S}}\nolimits _{\Psi }$. It is also shown, in particular, that (i) for any smooth $\mathop {\mathfrak {S}}\nolimits _{\Psi }$-field K any smooth finitely generated K-semilinear representation of $\mathop {\mathfrak {S}}\nolimits _{\Psi }$ is noetherian, (ii) for any $\mathop {\mathfrak {S}}\nolimits _{\Psi }$-invariant subfield K in the field $k(\Psi )$, the object $k(\Psi )$ is an injective cogenerator of the category of smooth K-semilinear representations of $\mathop {\mathfrak {S}}\nolimits _{\Psi }$, (iii) if $K\subset k(\Psi )$ is the subfield of rational homogeneous functions of degree 0 then there is a one-dimensional K-semilinear representation of $\mathop {\mathfrak {S}}\nolimits _{\Psi }$, whose integral tensor powers form a system of injective cogenerators of the category of smooth K-semilinear representations of $\mathop {\mathfrak {S}}\nolimits _{\Psi }$, (iv) if $K\subset k(\Psi )$ is the subfield generated over k by $x-y$ for all $x,y\in \Psi$ then there is a unique isomorphism class of indecomposable smooth K-semilinear representations of $\mathop {\mathfrak {S}}\nolimits _{\Psi }$ of each given finite length. Appendix collects some results on smooth linear representations of symmetric groups and of the automorphism group of an infinite-dimensional vector space over a finite field.