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{\large Groups, Geometry and Combinatorics }
\bigskip
{\em University of Durham, Durham, UK }
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{\bf July 16 -- 26, 2001}
\bigskip
{\Large A B S T R A C T S}
\bigskip
\bigskip
\end{center}
\begin{center}
{\em Rosemary A. Bailey}
\bigskip
{\bf Generalized wreath products of association schemes}
\end{center}
\noindent
The operation of forming the direct product of two association schemes
is commutative but the wreath product is not. Thus these operations
can be thought of as defined by the 2-point antichain and the 2-point
chain respectively. This can be generalized to any finite poset,
mirroring what can be done for permuation groups.
\bigskip
\begin{center}
{\em Alexander Baranov}
\bigskip
{\bf Plain representations of non-semisimple Lie algebras and
groups}
\end{center}
\noindent
Jointly with A. Zalesskii we started to study
representations of non-semisimple Lie algebras and algebraic (or finite)
groups.
One of the most important results obtained is the correspondence between
so-called ``plain'' Lie algebras and groups and finite dimensional
associative algebras. Moreover, it has been shown that ``plain''
representation
theory is almost equivalent to representation
theory of finite dimensional associative algebras.
Although the representation theory of
finite dimensional non-semisimple associative algebras is quite
developed,
there were no similar results for non-semisimple Lie algebras.
\bigskip
\begin{center}
{\em Barbara Baumeister}
\bigskip
{\bf Affine dual polar spaces}
\end{center}
\noindent
There is a uniform approach to consider interesting flag-transitive
geometries belonging to diagrams close to those of spherical buildings
-- interesting in the sense that they include many geometies with sporadic
automorphism group: The affinization of spherical buildings.
In my talk I will focus on the affinization of dual polar spaces.
They provide nice examples and it seems to be feasible to classify
them.
\bigskip
\begin{center}
{\em Jon Brundan}
\bigskip
{\bf Modular branching rules for the double covers of the symmetric groups}
\end{center}
\noindent
Recently I. Grojnowski has shown in detail how to connect the representation theory
of affine Hecke algebras to the Kac-Moody algebra of type $A_{p-1}^{(1)}$. In this
talk I will describe the analogues of these results for the twisted Kac-Moody algebra
of type $A_{p-1}^{(2)}$. One needs to replace the affine Hecke algebra with the affine
Hecke-Clifford superalgebra introduced by Jones and Nazarov. As a consequence, one
gets modular branching rules for the double covers of the symmetric group, analogous
to Kleshchev's modular branching rules for $S_n$ itself.
\bigskip
\begin{center}
{\em Peter J. Cameron}
\bigskip
{\bf Association schemes and permutation groups}
\end{center}
\noindent
Every permutation group gives rise in a natural way to a coherent
configuration. However, not every permutation group preserves a
non-trivial association scheme (a symmetric coherent configuration),
and it is possible that a group preserves some association scheme but
not a minimal association scheme with respect to refinement. I will
give some recent results about these conditions and their negations,
and how they relate to more familiar concepts from permutation group
theory and statistical design such as generous transitivity and
stratifiability. This is joint work with P.~Alejandro and R.~A.~Bailey.
\begin{center}
{\em Arjeh Cohen}
\bigskip
{\bf The LUP algorithm for other algebraic groups }
\end{center}
\noindent
The classical LUP algorithm decomposes a matrix into a product $LUP$
of a Lower triangular matrix $L$, an Upper triangular matrix $U$ and a
Permutation matrix $P$. The matrix $U$ can be chosen so that
$PUP^{-1} $ is lower triangular.
If the input matrix $A$ is invertible, the decomposition can be found
by elementary row and column operations on $A$, thereby keeping the
column operations to a minimum. The result is then a triple $K$, $N$,
$M$ of two lower triangular matrices $K$ and $M$ and a monomial matrix
$N$ (that is, $N= HP$, the product of diagonal matrix $H$ and a
permutation matrix $P$) such that $A = KNM$ and $NMN^{-1}$ is upper
triangular. The triple $L = KH$, $U = PMP^{-1}$, $P$ is then an LUP
decomposition of $A$. The decomposition $A=KNM$ is the so-called
Bruhat decomposition of an element of the general linear group.
In this context the question arises whether in any linear (highest
weight) representation of an algebraic group we can find the Bruhat
decomposition of a group element given by a matrix in the
representation. In the talk we shall address this question and
interpret the above LUP decomposition as a special case. The work
is joint with Scott Murray and Don Taylor.
\begin{center}
{\em Robert T. Curtis}
\bigskip
{\bf Symmetric generation: the way forward}
\end{center}
\noindent
A {\em progenitor} is defined to be a semi-direct product of
a free product of isomorphic cyclic groups by a group of monomial
automorphisms. Such a group is written $m^{*n}:N$, where $m^{*n}$
denotes a free copy of $n$ copies of the cyclic group of order $m$,
and $N$ acts on this free product by permuting a set of generators
for the cyclic groups and raising them to powers co-prime to $m$.
Thus a monomial representation of degree $n$ of the group $N$ over the
field $Z_p$ would enable us to define a progenitor of form $p^{*n}:N$.
Of course, when $m=2$ the group $N$ simply permutes the set of
$n$ involutions
which generate the cyclic groups.
It is easy to see that every simple group is a homomorphic image of
such progenitors (in many different ways), and over the last few years
considerable progress has been made in defining groups in this manner.
It turns out that the sporadic simple groups appear to be particularly
amenable to this approach, and factoring a suitable progenitor by just one
further short relator is usually enough to define the required image.
This approach is closely related to the amalgam methods exploited by
Parker and Rowley.
The smaller sporadic groups, including the Mathieu groups $J_1$,
$J_2$, $J_3$,
and $HS$, can be defined and constructed by hand in this way. Other groups,
such as the Held group and the Harada-Norton group, at present require
computation to verify that a certain progenitor factored by a single rather
natural relator yields the desired group. For the largest sporadic groups,
including the Monster, suitable progenitors are known, and relators exist
which are conjectured to be sufficient to define the group. However,
mechanical verification of these conjectures will remain out of range of
computers for the foreseeable future.
It is anticipated that modern geometric techniques, and in particular the approach
made famous by Conway, Ivanov and Norton in connection with $Y$-diagrams and the
Monster, will provide the way forward in proving these conjectures by hand
for all the outstanding cases.
\bigskip
\begin{center}
{\em Persi Diaconis}
\bigskip
{\bf The mathematics of shuffling cards}
\end{center}
\noindent
The well known result that seven shuffles suffice to mix up 52
cards has its roots in the descent theory of Coxeter groups.
There have been many refinements,
and extensions and applications; to buildings and hyperplane
arrangements,
cyclic homology, ergodic theory, symmetric function theory and fairly
applied
computer science. My plan is to review the results (old and new) in some
detail. There are also many lovely conjectures.
\bigskip
\begin{center}
{\em Peter Fleischmann }
\bigskip
{\bf Constructive modular invariant theory of finite groups }
\end{center}
\noindent
Let $G$ be a finite group, $V$ a finite dimensional $FG$- module over
the field $F$ and $A:=Sym(V^*)$ the symmetric algebra over the dual of
$V$. The ring of invariants $A^G:=\{ a \in A | ga = a {\rm ~for~all~}
g \in G \}$
is the main object of study in invariant theory. Classically the case where
F is the field of complex numbers and $G$ is a (possibly infinite) reductive
group has been studied and the theory here is well developed. In particular
it is known that $A^G$ is a Cohen - Macaulay ring which, in the finite group case,
is generated in degree less or equal to $|G|$
(called the `Noether bound').
New applications, e.g. in algebraic geometry and group cohomology,
ask for results about invariant rings of finite groups over
fields of characteristic $p>0$.
Here the theory is much less developed and has found new interest
during the last twenty years or so.
For example, if $p$ divides $|G|$, the invariant ring is in general not Cohen - Macaulay and
the computation of the depth of $A^G$ is a serious challenge.
Moreover Noether's degree bound also fails in this case and there is no replacement
known, except in special cases. Even if $0 < p$ does not divide $|G|$,
it has been a long standing conjecture that the Noether bound is
still valid, but this has been proved only quite recently.
In my talk I will give an account on recent results, current developments and
conjectures in modular invariant theory of finite groups. This includes
new interesting connections between the depth of $A^G$, the cohomology of G
and certain constructible `trace ideals' of $A^G$ and their varieties in the quotient
space.
\bigskip
\begin{center}
{\em Beth Holmes}
\bigskip
{\bf Computing in the Monster}
\end{center}
\noindent
Recent constructions of the Monster as implicit
$196882 \times 196882$ matrices over $GF(2)$ and $GF(3)$ have
led to effective computational capabilities and
opened up the possibility of solving several previously
intractable problems concerning this group.
For example, we have shown that the Monster is a
$(2,3,7)$-group, and that it contains a (previously
unknown) maximal subgroup $L_2(29):2$, as well as making
further inroads into the maximal subgroup problem.
\bigskip
\begin{center}
{\em Alexander Kleshchev}
\bigskip
{\bf Irreducible restrictions of linear
and projective representations of symmetric and alternating groups}
\end{center}
\noindent
We will report on the problem of describing the pairs $(G,D)$
where $G$ is a subgroup of $S_n$ (resp. $A_n$) and $D$ is a representation of
$S_n$ (resp. $A_n$), which is irreducible upon restriction to $G$.
This is of importance for the problem on maximal subgroups in finite classical
groups.
\bigskip
\begin{center}
{\em Ross Lawther}
\bigskip
{\bf Elements of specified order in simple algebraic groups}
\end{center}
\noindent
Let $\Phi$ be an irreducible root system, and $r$ be a natural number.
If $G$ is a simple algebraic group with root system $\Phi$, the variety of
elements $g\in G$ satisfying $g^r=1$ is considered. A lower bound for the
codimension in $G$ of this variety is obtained, which depends only on
$\Phi$ and $r$ (and in particular is independent of characteristic), and is
attained if $G$ is of adjoint type.
\bigskip
\begin{center}
{\em Caiheng Li}
\bigskip
{\bf The finite primitive permutation groups containing abelian
regular subgroups }
\end{center}
\noindent
In this talk, some results regarding permutation groups
containing regular subgroups and 2-arc-transitive Cayley graphs will be
reported:
A complete classification is given of finite primitive permutation
groups which contain abelian regular subgroups, solving a problem
initiated by Burnside (1911).
This classification is then applied to give a classification of
2-arc-transitive Cayley graphs of abelian groups, which was a recent
open problem in graph theory.
\bigskip
\begin{center}
{\em Alex Lubotzky}
\bigskip
{\bf Groups which are expanders and non expanders }
\end{center}
\noindent
In response to a question asked a decade ago by Weiss and the
speaker, we present an infinite family of finite groups whose Cayley
graphs are expanders w.r.t. to one chice of (bounded number of) generators
and are not w.r.t. another such chice. (Joint work with Avi Wigderson).
\bigskip
\begin{center}
{\em Kay Magaard}
\bigskip
{\bf Imprimitve representations of quasi-simple groups}
\end{center}
\noindent
Let $G$ be a quasi-simple finite group, $K$ an algebraically
closed field and $M$ an irreducible $KG$-module.
We say that $M$ is imprimitive with block stabilizer
$H \subset G$ if there exists some $KH$-module $M_0$ such that
$M= Ind_H^G(M_0)$. If no such $H$ exists we call $M$ a primitive
$KG$-module. Seitz proved that if $G$ is of Lie type of characteristic
$p$ and if $char(k) = p$, then with four exceptions
every irreducible $KG$-module is primitive. Djorkovic and
Malzan proved a similar result for characteristic zero modules
of alternating and symmetric groups.
I will present joint work with Gerhard Hiss and William Husen
that shows that the situation is very different when
$G$ is of Lie type of characteristic $p$ and if $char(K) \neq p$.
In fact in our case most irreducible $KG$-modules are
imprimitive. I will also discuss how our results fit into
the program of classifying maximal subgroups of classical groups.
\bigskip
\begin{center}
{\em Gunter Malle }
\bigskip
{\bf $2F$-modules}
\end{center}
\noindent
I report on joint work in progress with R.M.Guralnick
on the classification of $2F$-modules for quasi-simple groups and their
automorphism groups. These modules play a crucial role in the recent
second revision of parts of the classification by Meier-Frankenfeld
Stellmacher, Stroth et al.
\bigskip
\begin{center}
{\em Igor Pak}
\bigskip
{\bf The product replacement algorithm}
\end{center}
\noindent
I will survey recent progress on the
product replacement algorithm. The algorithm is
designed to generate random group elements. The results
cover connectivity, polynomial mixing and the universality
theorem.
\bigskip
\begin{center}
{\em Antonio Pasini}
\bigskip
{\bf Embeddings and expansions }
\end{center}
\noindent
Let $\cal G$ be a geometry belonging to a string diagram of rank
$n > 1$ where the nodes of the diagram are labelled by the integers
$1, 2,..., n$ from left to right, as usual. So, $\cal G$ can also be
regarded as a poset, where we write $x < y$ for two elements $x, y$
when $x$ and $y$ are incident and the type of $x$ is less then the type
of $y$. Denoted the set of 1-elements of $\cal G$ by $P$, I will write $P(x)$
for the set of 1-elements incident to a given element $x$. I also write
$x\in{\cal G}$ to say that $x$ is an element of $\cal G$. Having stated
these notations, we can define embeddings.
An {\em embedding} of $\cal G$ in a group $G$ is an injective mapping
$e$ from the poset $\cal G$ to the poset of non-trivial proper subgroups of
$G$ such that:
\begin{itemize}
\item[(1)] $x\leq y$ iff $e(x)\leq e(y)$, for any two $x, y\in{\cal G}$;
\item[(2)] $e(x) = \langle e(p)\rangle_{p\in P(x)}$ for any $x\in{\cal G}$;
\item[(3)] $G = \langle e(p)\rangle_{p\in P}$.
\end{itemize}
We denote by $A(e)$ the amalgam ${e(x)}_{x\in{\cal G}}$ and we define
the {\em expansion} $E(e)$ of $\cal G$ to $G$ via $e$ as the geometry
over the set of types $\{0, 1,..., n\}$ having the elements of $G$ as
0-elements and the right cosets of the subgroups $e(x)$ ($x\in{\cal G}$
of type $i$) as $i$-elements, the incidence relation being defined via
inclusion.
Clearly, denoted by $G(e)$ the universal completion of the amalgam $A(e)$,
an embedding $\tilde{e}$ is naturally given in $G(e)$. We call it the
{\em universal hull} of $e$. The following is the main result of my talk:
\paragraph{Theorem 1.} The expansion $E(\tilde{e})$ is the universal cover
of $E(e)$.
\bigskip
\noindent
Morphisms of embeddings can also be defined: if $e_1:{\cal G}\rightarrow G_1$
is another embedding of $\cal G$, a morphism $f:e_1\rightarrow e$ is a
homorphism $f:G_1\rightarrow G$ such that, for every $x\in{\cal G}$, $f$
induces on $e_1(x)$ an isomorphism to $e(x)$. (Note that $f_1$ induces a
covering from $E(e_1)$ to $E(e)$.) In particular, with $G(e)$ and $\tilde{e}$
as above, the natural projection $\pi:G(e)\rightarrow G$ is a morphism from
$\tilde{e}$ to $e$.
\paragraph{Proposition 2.} The pair $(\tilde{e}, \pi)$ is characterized
(up to isomorphism) by the following `universal' property: for every morphism
of embeddings $f:e_1\rightarrow e$, there exists a morphism
$g:\tilde{e}\rightarrow e_1$ such that $fg = \pi$.
\bigskip
\noindent
Finally, I will turn to projective embeddings (where $\cal G$ has rank 2, $G$
is the additive group of a vector space $V$ and the subgroups corresponding
to points and lines are 1- and 2-dimensional subspaces of $V$), revisiting
some known results and constructions on projective embeddings in the light
of the above general framework.
\bigskip
\begin{center}
{\em Cheryl Praeger }
\bigskip
{\bf Overgroups of finite quasiprimitive permutation groups }
\end{center}
\noindent
A finite transitive permutation group is quasiprimitive if each if its
non-trivial normal subgroups is transitive. Such groups often arise as
automorphism groups of combinatorial or geometric structures, and a
natural problem is to determine the full automorphism group. The group
theoretic equivalent is the problem of determining the overgroups of a
given quasiprimitive permutation group in the symmetric group. The heart
of this problem is to find all quasiprimitive overgroups. I will discuss
the extent to which this can be done in terms of the `O'Nan Scott type' of
a quasiprimitive group.
\bigskip
\begin{center}
{\em Alan Prince}
\bigskip
{\bf A search for a projective plane of order 20 }
\end{center}
\noindent
It is an open problem whether or not the order of a finite projective plane must be a prime-power. It was suggested by R. H. Bruck about 40 years ago that it might be possible to construct a projective plane of order $q(q+1)$, for some value of $q$, by extending the point-line geometry of PG(3,$q$). For certain values of $q$, such as $q=2$, the construction is ruled out by the Bruck-Ryser theorem. Of the remaining values, the construction has been shown to be impossible only for $q=3$. In this talk, I shall discuss a search for a projective plane of order 20, which extends the point-line geometry of PG(3,4).
\bigskip
\begin{center}
{\em Colva Roney-Dougal}
\bigskip
{\bf Affine Groups with Two Self-Paired
Orbitals }
\end{center}
\noindent
In this talk we define the {\em self-paired rank} of a group to be the
number of nondiagonal self-paired orbitals. We give a number of
reduction theorems for groups with self-paired rank 1, before
constructing some infinite families of soluble affine groups with
this property. We finish with an insoluble affine example.
\bigskip
\begin{center}
{\em Peter Rowley}
\bigskip
{\bf A Monster graph }
\end{center}
\noindent
This talk will be about the structure of
the point-line collinearity graph of the maximal 2-local geometry of
the Monster simple group.
\bigskip
\begin{center}
{\em Yoav Segev}
\bigskip
{\bf Finite quotients of the multiplicative group of a finite
dimensional division algebra are solvable}
\end{center}
\noindent
We prove the theorem of the title, using techniques developed
recently to obtain "congruence subgroup type" theorems over arbitrary
fields; and using a property of the commuting graph of minimal nonsolvable
groups (i.e., nonsolvable groups all of whose proper quotients are
solvable) which is stronger than having diameter $\ge 3$ and weaker than
having diameter $\ge 4$ (these graphs have diameter 3 and a half "so to
speak"). The theorem of the title is closely related to the
Margulis-Platonov conjecture for anisotropic algebraic groups of type
$A_n$. This work is joint with A.S. Rapinchuk and G.M. Seitz.
\bigskip
\begin{center}
{\em Johannes Siemons }
\bigskip
{\bf Representations from simplicial homology }
\end{center}
\noindent
Standard homology, defined via the usual
boundary map $\beta: [x_1,\, x_2,..,\,x_n]\mapsto
\sum\,\, (-1)^i \,\,[x_1,\,..,\,\hat{x_i},..,\,x_n]$, is
fundamental for understanding the topology of simplicial
complexes. However, there are other kinds of homology, and some
yield a rich representations theory for the groups associated
to such complexes. In the lecture I shall describe the homology
defined by the map $$\vspace{-0.3cm}\partial: \{x_1,\, x_2,..,\,x_n\}\mapsto
\sum_i\, \,\, \{x_1,\,..,\,\hat{x_i},..,\,x_n\}\,\,$$ and discuss
some of its properties in relation to standard homology.
\medskip
The first observation is that $\partial^p= 0$
if $p=0$ as an element of the coefficient domain. In particular, if
coefficients are taken from a field of characteristic $p>0$
then homology modules may be defined as quotients of
${\rm Ker} \,\partial ^{i}$ over ${\rm Im}\,\partial ^{p-i}$ with regards
to modules belonging to a suitably arranged sequence.
Already the ordinary $n$-simplex shows interesting
features: Its homology modules are trivial in all but
one position of every such sequence.
The non-trivial homology yields irreducible
${\rm Sym}(n)$-modules and these form the simplest
inductive systems of modular representations for the
symmetric groups. In addition, standard techniques
from algebraic topology such as the Hopf-Lefschetz
trace theorem give ready character formulae
and we shall discuss applications to orbits of
permutation groups whose order is co-prime to $p$.
\medskip
Having vanishing homology in all but one position
is an important property. For instance, the order complex of every
Cohen-Macauley poset has this property and there the non-trivial
homology is commonly regarded as the Steinberg representation
of the poset. In general the modular homology of such order
complexes turns out to be much richer. Nevertheless, certain
constructions carry over from ordinary homology and so
we have a reasonably detailed description for the modular homology
of all simplicial complexes which can be built up inductively from
spheres. These include Coxeter complexes and finite buidings.
For more details please go to my homepage
http://www.mth.uea.ac.uk/$\tilde{\phantom{.}}$h260/\,\,.
\bigskip
\begin{center}
{\em Steve Smith}
\bigskip
{\bf Final steps in the classification of quasithin groups }
\end{center}
\noindent
Aschbacher and Smith are revising the final two chapters
of their work on quasithin groups. The talk will indicate some
of the ideas used in the "endgame" cases of the work, such as
a new version of large-extraspecial theory, and some ideas
related to $N$-group type configurations.
\bigskip
\begin{center}
{\em Leonard Soicher}
\bigskip
{\bf Computing cliques in $G$-graphs, with applications to
finite geometry}
\end{center}
\noindent
A $G$-{\it graph} is a triple $(\Gamma,G,\psi)$, where $\Gamma$
is a graph, $G$ is a group, and $\psi$ is a homomorphism from $G$
into Aut$(\Gamma)$. Many classification problems in finite geometry
reduce to the problem of classifying certain $G$-orbits of cliques
in a given $G$-graph. I will talk about some advances in search
algorithms for computing representatives of the $G$-orbits of cliques
(possibly satisfying certain conditions) in a $G$-graph, as well as
some applications of these algorithms to problems in finite geometry.
The first problem is the classification of maximal partial spreads
$P$ of lines in a given projective space $PG(n,q)$, such that $P$ is
invariant under a given subgroup of $PGL(n+1,q)$. The second problem
is the discovery and classification of certain SOMA$(k,n)$s, which are a
class of combinatorial designs generalizing the concept of $k$ mutually
orthogonal Latin squares of order $n$.
\bigskip
\begin{center}
{\em Irina D. Suprunenko }
\bigskip
{\bf Properties of unipotent elements in modular irreducible\\
representations of the classical algebraic groups}
\end{center}
\noindent
Some properties of unipotent elements in modular irreducible
representations of the classical algebraic groups specific for positive
characteristic are discussed. The bulk of the talk will be concerned
with
computing the minimal polynomials of the images of unipotent elements of
nonprime order in irreducible representations of the classical algebraic
groups
in odd characteristic. Actually, for a fixed such element of order $p^s$
and
a fixed representation of a classical algebraic group in characteristic
$p>2$
the problem is reduced to computing the minimal polynomials of $s$
certain
unipotent elements in a certain irreducible representation of a certain
semisimple algebraic group in characteristic 0 with classical simple
components. For the latter question explicit algorithms are available.
Some other problems on the behavior of unipotent elements in
representations
can be considered as well (this depends upon the length of my talk).
These
problems are connected with the number of Jordan blocks of the maximal
size
in the image of a unipotent element in an irreducible representation
with
highest weight large enough with respect to the (positive)
characteristic
and with recognizing the classical algebraic groups with the help of
matrices
with big Jordan blocks with respect to the degree of a group.
\bigskip
\begin{center}
{\em Balas Szegedy}
\bigskip
{\bf On the characters of the Borel and Sylow subgroups of classical groups}
\end{center}
\noindent
M. Isaacs has proved that every irreducible complex character of the
group of all upper unitriangular $GF(q)$ matrices has $q$-power degree.
The analogous question for the Sylow subgroups of all classical groups
will be discussed. These results have also important consequences for
the Borel subgroups.
\bigskip
\begin{center}
{\em Paul Terwilliger }
\bigskip
{\bf Leonard pairs associated with the classical geometries }
\end{center}
\noindent
Let $V$ denote a complex vector space with finite
positive dimension, and let $A:V \to V$ and
$B:V \to V$ denote linear transformations.
The pair $A, B$ is said to be a {\it Leonard pair }
whenever both conditions (i), (ii) hold below:
\begin{enumerate}
\item There exists a basis for $V$ with respect to which
the matrix representing $A$ is diagonal, and the matrix
representing $B$ is irreducible tridiagonal.
\item There exists a basis for $V$ with respect to which
the matrix representing $B$ is diagonal, and the matrix
representing $A$ is irreducible tridiagonal.
\end{enumerate}
It is known that Leonard pairs are closely related
to the $q$-Racah polynomials and related polynomials
from the Askey Scheme. In this talk, we show how
Leonard pairs are naturally related to the
following classical geometries:
The subset lattice,
the subspace lattice, the Hamming semi-lattice,
the attenuated spaces, and the classical polar spaces.
\bigskip
\begin{center}
{\em Pham Huu Tiep}
\bigskip
{\bf Cross-characteristic representations of finite groups of Lie type}
\end{center}
\noindent
Let $G$ be a finite group of Lie type defined in characteristic $p$. Then the
degree of any nontrivial irreducible (projective) representation of $G$ in
characteristic $\neq p$ is bounded below by the Landazuri-Seitz-Zalesskii
bound, $\ell_{LSZ}(G)$. In a number of applications it is important to
know all the cross-characteristic representations of $G$ of degree less than
$(\ell_{LSZ}(G))^{2-\epsilon}$. This problem has been solved for $G = SL_{n}(q)$
by R. M. Guralnick and P. H. Tiep, and for $G = SU_{n}(q)$ by G. Hiss and
G. Malle. We solve the problem for the case $G = Sp_{2n}(q)$ with $q$ odd. We
also improve the results of Hiss and Malle for the case $G = SU_{n}(q)$.
Applications of our results to the minimal polynomial problem, the
quadratic pair problem and to submodule structure of various rank 3
permutation modules will also be given. This talk is mostly based on joint
work with R. M. Guralnick, K. Magaard and J. Saxl.
\bigskip
\begin{center}
{\em Franz Timmesfeld}
\bigskip
{\bf Classifications of Lie-type groups}
\end{center}
\noindent
Let $\cal {B}$ be an irreducible spherical building, $\cal{A}$ an
apartment of $\cal{B}$. Then the subgroup of ${\rm Aut}(\cal{B})$
generated by a root-subgroup corresponding to roots in $\cal{A}$,
will be called the group of Lie-type $\cal{B}$.
I will speak about general classifications of such groups of Lie-type
$\cal{B}$. For example a classification of groups with a `` local
BN-pair ``, which generalizes the notion of split BN-pair.
\bigskip
\begin{center}
{\em Vladimir I. Trofimov}
\bigskip
{\bf Track method for amalgams with groups of Lie type}
\end{center}
\noindent
The track method of investigation of amalgams with groups
of Lie type (including $PSL_n(q))$ is set forth.
A brief survey of results obtained by the method
is also given, and perspectives are discussed.
\bigskip
\begin{center}
{\em Robert A. Wilson }
\bigskip
{\bf Black box algorithms for
recognising simple groups}
\end{center}
\noindent
Algorithms for black box groups have been developed
over a number of years to the point where Monte Carlo
polynomial time algorithms exist for many problems,
in particular for recognising a group if it is
know to be simple. The problem of recognising a simple
group if it is not known in advance to be simple
is much harder, the main stumbling block being to
distinguich between a group of Lie type in characteristic $p$,
and the same group acting on a module in defining characteristic.
We solve this problem for $p$ odd.
\bigskip
\end{document}
A change of plan concerning my talk:
Beth Holmes will talk on "Computing in the Monster"
(tha abstract I wrote will do for her talk),
and I will talk on ""
Abstract:
Rob.
--
Professor Robert A. Wilson
School of Mathematics and Statistics, The University of Birmingham,
Edgbaston, Birmingham B15 2TT, U.K.
email: R.A.Wilson@bham.ac.uk tel: +44-121-414-6605 fax: +44-121-414-3389