Advertisement

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
Article
  • 30 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.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Bucur, I., Deleanu, A.: Introduction to the Theory of Categories and Functors. Wiley, Hoboken (1968)zbMATHGoogle Scholar
  2. 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. 3.
    MacLane, S.: The universality of formal power series fields. Bull. Am. Math. Soc. 45(12), 888–890 (1939)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Neumann, B.H.: Groups covered by permutable subsets. J. Lond. Math. Soc. 29(2), 236–248 (1954)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Rovinsky, M.: Semilinear representations of PGL. Sel. Math. New Ser. 11(3–4), 491–522 (2005). arXiv:math/0306333 MathSciNetzbMATHGoogle Scholar
  6. 6.
    Rovinsky, M.: Admissible semi-linear representations. J. Reine Angew. Math. (Crelle) 604, 159–186 (2007). arXiv:math.RT/0506043 MathSciNetzbMATHGoogle Scholar
  7. 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. 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. 9.
    Rovinsky, M.: Stable birational invariants with Galois descent and differential forms. Mosc. Math. J. 803(4), 777 (2015)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Serre, J.-P.: Corps locaux, 3rd edn. Hermann, Houston (1968)zbMATHGoogle Scholar
  11. 11.
    Speiser, A.: Zahlentheoretische Sätze aus der Gruppentheorie. Math. Zeit. 5(1/2), 1–6 (1919)CrossRefzbMATHGoogle Scholar

Copyright information

© 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

Personalised recommendations