“Norman involutions” and tensor products of unipotent Jordan blocks

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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) = Sa ≀Db 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.

Original languageEnglish
Pages (from-to)153-181
JournalIsrael Journal of Mathematics
Volume230
Issue number1
Early online date6 Dec 2018
DOIs
Publication statusPublished - Mar 2019

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