“Norman involutions” and tensor products of unipotent Jordan blocks

Research output: Contribution to journalArticle

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

Fingerprint

Jordan Canonical Form
Jordan Block
Involution
Tensor Product
Permutation
Dihedral group
Wreath Product
Reversal
Symmetric group
Factorization
Trivial
Partition
Necessary Conditions
Sufficient Conditions

Cite this

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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.",
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“Norman involutions” and tensor products of unipotent Jordan blocks. / Glasby, S. P.; Praeger, Cheryl E.; Xia, Binzhou.

In: Israel Journal of Mathematics, Vol. 230, No. 1, 03.2019, p. 153-181.

Research output: Contribution to journalArticle

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AB - 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.

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