# Splinter Group:

Group and Representation Theory

Organizer: Kevin Jennings (NUI Galway)
All talks will be held in the

**Dillon Lecture Theatre**.

## Wednesday, April 8 2009

14:15 - 14:45

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James Shank (University of Kent):

A first main theorem for SL_2(\Bbb F_p)

I will describe the construction of a generating set for the ring of vector invariants \Bbb F[mV]^{SL_2(\Bbb F_p)}, where \Bbb F is any field of characteristic p and V is the defining representation of SL_2(\Bbb F_p). This is joint work with Eddy Campbell(Memorial) and David Wehlau(RMC).

14:45 - 15:15

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Ben Wright (The University of Manchester):

The Structure of a Point Line Collinearity Graph for Fi_{24}'

This talk presents joint work With Peter Rowley and Louise Walker on a certain graph associated with the largest simple Fischer group Fi_{24}^{\prime}. This graph is the
point-line collinearity graph of the maximal 2-local geometry which was first introduced by
Ronan and Smith in 1981 Santa Cruz Proceedings.
A very detailed picture has been obtained of this graph which has 2,503,413,946,215 vertices and 120 suborbits.

15:15 - 15:45

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A. G. O'Farrell (NUI Maynooth):

Reversibility

An element of a group is said to be reversible if it is conjugate to its inverse. The notion originated in dynamics, and arises in many other problems. We will discuss some progress on the problem of identifying the reversible elements of specific (large) groups.

15:45 - 16:15

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Paul Taylor (University of Manchester):

Computing normalizers of 2-groups in black-box groups

We present an extension of an algorithm in a paper of Bates and Rowley for computing the normalizer of a 2-subgroup of a black-box group, with particular focus on the case where the 2-subgroup is elementary abelian.

16:30 - 17:00

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Yuri Bazlov (University of Warwick):

Even elements in complex reflection groups and noncommutative Dunkl operators

Complex reflection groups are a remarkable class of finite linear groups, characterised by polynomiality of their ring of invariants. Irreducible groups of this class are split into three infinite "classical" series plus 34 exceptional groups. To each complex reflection group is associated a set of commuting Dunkl operators. I will explain how to introduce anticommuting Dunkl operators (using the braided doubles technique due to Berenstein and myself), which leads to a new "braided" family of groups. They include certain subgroups of index 2 (subgroups of even elements) in classical complex reflection groups.