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Regular version of the site

Article

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

Selecta Mathematica, New Series. 2018. Vol. 24. No. 3. P. 2319-2349.

Let $K$ be a field and $G$ be a group of its automorphisms endowed with the compact-open topology, cf. section 1.1.

If $G$ is precompact then $K$ is a generator of the category of {\sl smooth} (i.e. with open stabilizers) $K$-{\sl 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 $\Sy_{\Psi}$ 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 $\Sy_{\Psi}$-action). One of principal results describes the Gabriel spectrum of the category of smooth $k(\Psi)$-semilinear representations of $\Sy_{\Psi}$.

It is also shown, in particular, that (i) for any smooth $\Sy_{\Psi}$-field $K$ any smooth finitely generated $K$-semilinear representation of $\Sy_{\Psi}$ is noetherian,  (ii) for any $\Sy_{\Psi}$-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 $\Sy_{\Psi}$, (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 $\Sy_{\Psi}$, whose integral tensor powers form a system of injective cogenerators of the category of smooth $K$-semilinear representations of $\Sy_{\Psi}$, (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 $\Sy_{\Psi}$ of each given finite length.

Appendix collects some results on smooth {\sl linear} representations of symmetric groups and of the automorphism group of an infinite-dimensional vector space over a finite field.