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“Norman involutions” and tensor products of unipotent Jordan blocks

  • S. P. Glasby
  • Cheryl E. Praeger
  • Binzhou Xia
Article

Abstract

This paper studies the Jordan canonical form (JCF) of the tensor product of two unipotent Jordan blocks over a field of prime characteristic p. The JCF is characterized by a partition λ = λ(r, s, p) depending on the dimensions r, s of the Jordan blocks, and on p. Equivalently, we study a permutation π = π(r, s, p) of {1, 2,..., r} introduced by Norman. We show that π(r, s, p) is an involution involving reversals, or is the identity permutation. We prove that the group G(r, p) generated by π(r, s, p) for all s, “factors” as a wreath product corresponding to the factorisation r = ab as a product of its p′-part a and p-part b: precisely G(r, p) = SaDb where Sa is a symmetric group of degree a, and Db is a dihedral group of degree b. We also give simple necessary and sufficient conditions for π(r, s, p) to be trivial.

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Copyright information

© Hebrew University of Jerusalem 2018

Authors and Affiliations

  • S. P. Glasby
    • 1
  • Cheryl E. Praeger
    • 1
  • Binzhou Xia
    • 1
  1. 1.Centre for Mathematics of Symmetry and ComputationUniversity of Western AustraliaCrawleyAustralia

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