# “Norman involutions” and tensor products of unipotent Jordan blocks

- 7 Downloads

## 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.

## Preview

Unable to display preview. Download preview PDF.

## References

- [1]A. C. Aitken,
*The normal form of compound and induced matrices*, Proceedings of the London Mathematical Society**38**(1934), 353–376.MathSciNetzbMATHGoogle Scholar - [2]M. J. J. Barry,
*On a question of Glasby*, Praeger, and Xia, Communications in Algebra**43**(2015), 4231–4246.MathSciNetCrossRefGoogle Scholar - [3]S. P. Glasby, Magma, code available at http://www.maths.uwa.edu.au/~glasby/research.html.Google Scholar
- [4]S. P. Glasby, C. E. Praeger and B. Xia,
*Decomposing modular tensor products: “Jordan partitions” their parts and p-parts*, Israel Journal of Mathematics**209**(2015), 215–233.MathSciNetCrossRefGoogle Scholar - [5]S. P. Glasby, C. E. Praeger and B. Xia,
*Decomposing modular tensor products, and periodicity of “Jordan partitions”*, Journal of Algebra**450**(2016), 570–587.MathSciNetCrossRefGoogle Scholar - [6]J. A. Green,
*The modular representation algebra of a finite group*, Illinois Journal of Mathematics**6**(1962), 607–619.MathSciNetzbMATHGoogle Scholar - [7]K.-I. Iima and R. Iwamatsu,
*On the Jordan decomposition of tensored matrices of Jordan canonical forms*, Mathematical Journal of Okayama University**51**(2009), 133–148.MathSciNetzbMATHGoogle Scholar - [8]R. Lawther,
*Jordan block sizes of unipotent elements in exceptional algebraic groups*, Communications in Algebra**23**(1995), 4125–4156.MathSciNetCrossRefGoogle Scholar - [9]R. Lawther,
*Correction to: “Jordan block sizes of unipotent elements in exceptional algebraic groups” [Comm. Algebra 23 (1995), no. 11, 4125–4156; MR1351124 (96h:20084)]*, Communications in Algebra**26**(1998), 2709.MathSciNetGoogle Scholar - [10]C. W. Norman,
*On the Jordan form of the tensor product over fields of prime characteristic*, Linear and Multilinear Algebra**38**(1995), 351–371.MathSciNetCrossRefGoogle Scholar - [11]J.-C. Renaud,
*The decomposition of products in the modular representation ring of a cyclic group of prime power order*, Journal of Algebra**58**(1979), 1–11.MathSciNetCrossRefGoogle Scholar - [12]P. C. Roberts, A computation of local cohomology,
*in Commutative Algebra: Syzygies, Multiplicities, and Birational Algebra (South Hadley, MA, 1992), Contemporary Mathematics, Vol. 159, American Mathematical Society*, Providence, RI, 1994, pp. 351–356.Google Scholar - [13]B. Srinivasan,
*The modular representation ring of a cyclic p-group*, Proceedings of the London Mathematical Society**14**(1964), 677–688.MathSciNetCrossRefGoogle Scholar