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Growth types of Coxeter groups and their quotientsAlgebra & Discrete Mathematics
|Speaker: ||Viswanath Sankaran, UC Davis|
|Location: ||1147 MSB|
|Start time: ||Thu, May 11 2006, 12:10PM|
Coxeter groups are a very special class of groups defined via
generators and relations. They are intimately related to the classical
finite dimensional semisimple Lie algebras and to their infinite
dimensional counterparts - the affine and indefinite Kac-Moody
algebras. In this talk, we focus on questions related to the "growth type"
of such groups; roughly, if gamma(n) denotes the number of elements
of the group of length <=n, the growth type measures the rate of
growth of gamma(n) with n.
It is well known that Coxeter groups of finite and affine type have
polynomial growth while all other Coxeter groups have exponential
growth. I will describe a generalization of this latter result to
quotients $W/W_J$ of such Coxeter groups by their parabolic subgroups. Along the way, we will introduce reflection subgroups of $W$ and
a criterion of M. Dyer in terms of inner products of roots that
enables construction of such subgroups. We'll use this and some root
system combinatorics to construct certain reflection subgroups of $W$
that are isomorphic to "universal Coxeter groups". This will lead us
to the result that the quotients $W/W_J$ have exponential growth too.