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Working paper

On semilinear representations of the infinite symmetric group

In this note the smooth (i.e. with open stabilizers) linear and {\sl semilinear} representations of certain permutation groups (such as infinite symmetric group or automorphism group of an infinite-dimensional vector space over a finite field) are studied. Many results here are well-known to the experts, at least in the case of {\sl linear representations} of symmetric group. The presented results suggest, in particular, an analogue of Hilbert's Theorem 90 should hold: in the case of faithful action of the group on the base field the irreducible semilinear representations are one-dimensional (and trivial in appropriate sense).