Page 67 - Textos de Matemática Vol. 44

A PASCAL-TYPE MATRIX 57
Remark
Several algorithms are known for computing the Jordan canonical form of Jm ⊗ Jn over general fields, as well as the structure of exterior powers and symmetric powers, and some references are given below. For example, the authors’ paper [2] deals with exterior squares, while the paper of Barry, [1], treats a wider range of problems.
The main interest in the current note is that the information is encoded in the LQU decomposition of the Pascal-type matrix Pn,m and it “explains” the fact that the permutation matrix Q occurring in the LQU decomposition over the integers modulo a prime p of the matrix Pn,m varies so much as one varies p.
Acknowledgement
We wish to thank the referee for very carefully reading the first version of this paper and identifying a number of inaccuracies.
References
[1] M. J. J. Barry, Decomposing tensor products and exterior and symmetric squares, J. Group Theory 14 (2011), 59–82.
[2] R. Gow and T. J. Laffey, On the decomposition of the exterior square of an indecompos- able module of a cyclic p-group, J. Group Theory 9 (2006), 659–672.
[3] J. H. Lindsey, II, Groups with a T. I. cyclic Sylow subgroup, J. Algebra 30 (1974), 181– 235.
[4] C. W. Norman, On the Jordan form of the tensor product over fields of prime charac- teristic, Linear and Multilinear Algebra 38 (1995), 351–371.
[5] J.-C. Renaud, The decomposition of products in the modular representation ring of a cyclic group of prime power order, J. Algebra 58 (1979), 1–11.
[6] B. Srinivasan, The modular representation ring of a cyclic p-group, Proc. London Math. Soc. 14 (1964), 677–688.
(R. Gow and T. J. Laffey) School of Mathematical Sciences, University College, Belfield, Dublin 4, Ireland
E-mail address: rod.gow@ucd.ie, thomas.laffey@ucd.ie