Selecta Mathematica

, Volume 24, Issue 3, pp 2319–2349 | Cite as

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

  • M. RovinskyEmail author


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 


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© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Laboratory of Algebraic GeometryNational Research University Higher School of EconomicsMoscowRussia
  2. 2.Institute for Information Transmission Problems of Russian Academy of SciencesMoscowRussia

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