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

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## 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.

## Mathematics Subject Classification

20C32 16S35 16D90 18F20 14C15## Preview

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