SEMINAR: Groups and Combinatorics Seminar:Serendipity, involutions and regular semisimple matrices
|Groups and Combinatorics Seminar:Serendipity, involutions and regular semisimple matrices
Groups and Combinatorics Seminar
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.
Maths Lecture Room 2
Tue, 20 Apr 2010 12:00
Tue, 20 Apr 2010 12:45
Michael Giudici <[email protected]>
Thu, 15 Apr 2010 17:09
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