Affine Weyl groups, grids and coloured tableaux
Ronald C. King *
School of Mathematics
University of Southampton
Southampton, SO17 1BJ
England
Abstract Full Text PDF
The Kac-Moody classification of affine Lie algebras includes the seven infinite series
and
. Let
be any one of these algebras and let
be the corresponding affine Weyl group, with generators
for
It is shown that the action of any element
of the Weyl group on any weight vector
can be codified by means of periodic grids, one for each
.
These grids are such that
is specified by a coloured tableaux
whose shape in the natural ε-basis is determined by
where
is the Weyl vector, and whose entries are the components of
in the basis of fundamental weights. At the heart of this calculation of
is a recursion such
that if
then
is obtained from
by using the
periodic grid of
to encode the required calculation of
.
Each affine Lie algebra
contains a maximal finite-dimensional simple Lie subalgebra
. Let
be the Weyl group of and let
be a set of minimal right coset representatives of
with respect to
. It is shown that if 
then
if and only if
has row lengths specified by the components of a dominant weight vector of
in the ε-basis. The bijective correspondence between
and
is then used to specify canonical expressions for the elements of
as repeated products of certain words in the generators
of
for each of the classical affine Lie algebras
.
The above results are then used to write the character of an arbitrary integrable highest weight irreducible representation
of
in terms of characters of irreducible representations of
.
* This is joint work with Trevor A. Welsh.
