‘Norman involutions’ and tensor products of unipotent Jordan blocks

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) = S_a ≀ D_b where S_a is a symmetric group of degree a, and D_b is a dihedral group of degree b. We also give simple necessary and sufficient conditions for π(r, s, p) to be trivial.