Resumen de Semilinear representations of symmetric groups and of automorphism groups of universal domains
M. Rovinsky
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 SΨ of all permutations of an infinite set Ψ are studied. Let k be a field and k(Ψ) be the field freely generated over k by the set Ψ (endowed with the natural SΨ -action). One of principal results describes the Gabriel spectrum of the category of smooth k(Ψ) -semilinear representations of SΨ . It is also shown, in particular, that (i) for any smooth SΨ -field K any smooth finitely generated K-semilinear representation of SΨ is noetherian, (ii) for any SΨ -invariant subfield K in the field k(Ψ) , the object k(Ψ) is an injective cogenerator of the category of smooth K-semilinear representations of SΨ , (iii) if K⊂k(Ψ) is the subfield of rational homogeneous functions of degree 0 then there is a one-dimensional K-semilinear representation of SΨ , whose integral tensor powers form a system of injective cogenerators of the category of smooth K-semilinear representations of SΨ , (iv) if K⊂k(Ψ) is the subfield generated over k by x−y for all x,y∈Ψ then there is a unique isomorphism class of indecomposable smooth K-semilinear representations of SΨ 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.
