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SEMINAR: Groups and Combinatorics Seminar:Serendipity, involutions and regular semisimple matrices

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Groups and Combinatorics Seminar

Cheryl Praeger

will speak on

Serendipity, involutions and regular semisimple matrices

at 12noon in MLR2 on Tuesday April 20.

Abstract: (joint work with Akos Seress)

Key to studying finite simple groups is to study their involution centralisers, and this is true also in a computational setting. In a seminal paper, Chris Parker and Rob Wilson present and analyse a practical algorithm for computing the centraliser of an involution z in a finite classical group of odd characteristic. Fundamental to their approach, using Bray’s theorem and an observation of Richard Parker, is the estimation of the proportion of pairs of conjugates of z whose product has odd order -- and is regular semisimple. Their analysis estimates that O(n) random selections are needed to find a suitable random conjugate of z.

Motivated by the wish to improve on Parker and Wilson's algorithm by exploiting the pairs of conjugates of z whose product has even order, we were led by experimental evidence to estimate the proportion of such pairs whose product is regular semisimple. Serendipity came to our rescue: we recognised a probability generating function for estimating such pairs as similar to one analysed by Jason Fulman, Peter Neumann and me for computing the proportion of separable matrices in unitary groups. A similar analysis enabled us find the limiting proportion, depending only on the field size q, for general linear groups and to prove exponentially fast convergence.

Speaker(s) Cheryl Praeger
Location Maths Lecture Room 2
Contact Michael Giudici <[email protected]>
Start Tue, 20 Apr 2010 12:00
End Tue, 20 Apr 2010 12:45
Submitted by Michael Giudici <[email protected]>
Last Updated Thu, 15 Apr 2010 17:09
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