# “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.

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