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Invariant fields of rational functions and semilinear representations of symmetric groups over them
Cornell University
,
2023.
No. 2205.15144.
Let K be a field and G be a group of its automorphisms endowed with the compact-open topology, cf. §1.1. There are many situations, where it is natural to study the category Sm_K(G) of smooth (i.e. with open stabilizers) K-semilinear representations of G. According to Hilbert’s Theorem 90, cf. Proposition 2.2, the category Sm_K(G) is semisimple (in which case K is a generator of Sm_K(G)) if and only if G is precompact. In this note we study the case of the non-precompact group G = S Ψ of all permutations of an infinite set Ψ. It is shown that the categories Sm K ( S Ψ ) are locally noetherian; the morphisms are ‘locally split’. Given a field F and a subfield k\neq F algebraically closed in F , one of principal results (Theorem 1.3) describes the Gabriel spectra (and related objects) of the categories Sm_K(S_Ψ) for some of S_Ψ -invariant subfields K of the fraction field F k,Ψ of the tensor product over k of the labeled by Ψ copies of F. In particular, the object F_{k,Ψ} turns out to be an injective cogenerator of the category Sm_K(S_Ψ) for any S Ψ -invariant subfield K ⊆ F_{k,Ψ}. As an application of Theorem 1.3, when transcendence degree of F |k is 1, a correspondence between the S Ψ -invariant subfields of F k,Ψ algebraically closed in F_{k,Ψ} and certain systems of isogenies of ‘generically F-pointed’ torsors over absolutely irreducible one-dimensional algebraic k-groups is constructed, so far only in characteristic 0. The only irreducible finite-dimensional smooth representation of S_Ψ is trivial. However, under some mild restriction on the field k, for each integer 1 ≤ n ≤ 5 it is possible to find an invariant subfield K of k(Ψ) and an irreducible non-trivial n-dimensional smooth K-semilinear representation of S_Ψ . It is likely that there are no such representations for n > 5. In general, any smooth S_Ψ -field K admits a smooth S Ψ -field extension L|K such that L is a cogenerator of Sm_L( S_Ψ ), cf. Proposition 4.29. Appendix collects some results on smooth characteristic 0 linear representations of symmetric groups and of the automorphism group of an infinite-dimensional vector space over a finite field.