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

- 31 Downloads

## Abstract

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## Preview

Unable to display preview. Download preview PDF.

## References

- 1.Bucur, I., Deleanu, A.: Introduction to the Theory of Categories and Functors. Wiley, Hoboken (1968)zbMATHGoogle Scholar
- 2.Gan, W.L., Li, L.: Noetherian property of infinite EI categories. N. Y. J. Math.
**21**, 369–382 (2015). arXiv:1407.8235v2 MathSciNetzbMATHGoogle Scholar - 3.MacLane, S.: The universality of formal power series fields. Bull. Am. Math. Soc.
**45**(12), 888–890 (1939)MathSciNetCrossRefzbMATHGoogle Scholar - 4.Neumann, B.H.: Groups covered by permutable subsets. J. Lond. Math. Soc.
**29**(2), 236–248 (1954)MathSciNetCrossRefzbMATHGoogle Scholar - 5.Rovinsky, M.: Semilinear representations of PGL. Sel. Math. New Ser.
**11**(3–4), 491–522 (2005). arXiv:math/0306333 MathSciNetzbMATHGoogle Scholar - 6.Rovinsky, M.: Admissible semi-linear representations. J. Reine Angew. Math. (Crelle)
**604**, 159–186 (2007). arXiv:math.RT/0506043 MathSciNetzbMATHGoogle Scholar - 7.Rovinsky, M.: Automorphism groups of fields, and their representations. Uspekhi Matem. Nauk
**62**(378) (2007), no. 6, 87–156, English translation Russ. Math. Surv.**62**(6), 1121–1186 (2007)Google Scholar - 8.Rovinsky, M.: On maximal proper subgroups of field automorphism groups. Sel. Math. New Ser.
**15**, 343–376 (2009). arXiv:math/0601028 MathSciNetCrossRefzbMATHGoogle Scholar - 9.Rovinsky, M.: Stable birational invariants with Galois descent and differential forms. Mosc. Math. J.
**803**(4), 777 (2015)MathSciNetzbMATHGoogle Scholar - 10.Serre, J.-P.: Corps locaux, 3rd edn. Hermann, Houston (1968)zbMATHGoogle Scholar
- 11.Speiser, A.: Zahlentheoretische Sätze aus der Gruppentheorie. Math. Zeit.
**5**(1/2), 1–6 (1919)CrossRefzbMATHGoogle Scholar