Signed Young Modules and Simple Specht Modules

Preprint English OPEN
Danz, Susanne; Lim, Kay Jin;
  • Subject: Mathematics - Representation Theory
    arxiv: Mathematics::Representation Theory | Astrophysics::Galaxy Astrophysics

By a result of Hemmer, every simple Specht module of a finite symmetric group over a field of odd characteristic is a signed Young module. While Specht modules are parametrized by partitions, indecomposable signed Young modules are parametrized by certain pairs of parti... View more
  • References (24)
    24 references, page 1 of 3

    2.7. Young subgroups, wreath products, Sylow p-subgroups of Sn. Let n ∈ Z+ . (a) For a partition, or, more generally, a composition λ = (λ1, . . . , λk) of n, we denote by Sλ = Sλ1 × · · · × Sλk the corresponding standard Young subgroup of Sn. (b) Let G be any finite group, and let H 6 Sn. We have the wreath product G ≀ H := {(g1, . . . , gn; σ) : g1, . . . , gn ∈ G, σ ∈ H}, whose multiplication is given by the trivial F Sα-module F (α) to Sk ≀ Sα. Moreover, if k is odd then s\gn(k) s\gn(k)⊗βt ∼= sgn(Sk ≀ Sβ) (see Remark 3.11). Now set [2] M. Broué, On Scott modules and p-permutation modules: an approach through the Brauer morphism, Proc. Amer. Math. Soc. 93 (1985), no. 3, 401-408.

    [4] S. Danz, On vertices of exterior powers of the natural simple module for the symmetric group in odd characteristic, Arch. Math. 89 (2007), no. 6, 485-496.

    [6] S. Danz, E. Giannelli, Vertices of simple modules of symmetric groups labelled by hook partitions, to appear in J. Group Theory, DOI 10.1515/jgth-2014-0044.

    [7] S. Danz, B. Külshammer, R. Zimmermann, On vertices of simple modules for symmetric groups of small degrees, J. Algebra 320 (2008), no. 2, 680-707.

    [8] S. Donkin, Symmetric and exterior powers, linear source modules and representations of Schur superalgebras, Proc. London Math. Soc. 83 (2001), 647-680.

    [9] K. Erdmann, Young modules for symmetric groups. Special issue on group theory, J. Aust. Math. Soc. 71 (2001), 201-210.

    [10] L. Evens, The cohomology of groups, Oxford Mathematical Monographs, Oxford University Press, New York, 1991.

    [11] M. Fayers, Reducible Specht modules, J. Algebra 280 (2004), no. 2, 500-504.

    [12] M. Fayers, Irreducible Specht modules for Hecke algebras of type A, Adv. Math. 193 (2005), 438-452.

    [22] G. James, The Representation Theory of the Symmetric Groups, Lecture Notes in Mathematics 682. Springer, Berlin, 1978.

  • Related Organizations (1)
  • Metrics
Share - Bookmark