### 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.

Original language | English |
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Pages (from-to) | 153-181 |

Journal | Israel Journal of Mathematics |

Volume | 230 |

Issue number | 1 |

Early online date | 6 Dec 2018 |

DOIs | |

Publication status | Published - Mar 2019 |

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*Israel Journal of Mathematics*, vol. 230, no. 1, pp. 153-181. https://doi.org/10.1007/s11856-018-1812-z

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

Research output: Contribution to journal › Article

TY - JOUR

T1 - “Norman involutions” and tensor products of unipotent Jordan blocks

AU - Glasby, S. P.

AU - Praeger, Cheryl E.

AU - Xia, Binzhou

PY - 2019/3

Y1 - 2019/3

N2 - 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.

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.

UR - http://www.scopus.com/inward/record.url?scp=85058058432&partnerID=8YFLogxK

U2 - 10.1007/s11856-018-1812-z

DO - 10.1007/s11856-018-1812-z

M3 - Article

VL - 230

SP - 153

EP - 181

JO - Israel Journal of Mathematics

JF - Israel Journal of Mathematics

SN - 0021-2172

IS - 1

ER -