Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
Last update: 26 June 2012
The affine Weyl group
This section is a summary of the main facts and notations that are needed for working with the affine Weyl group The main point is that the elements of the affine Weyl group can be identified with alcoves via the bijections in (2.11).
Let be a finite dimensional vector space over A reflection is a diagonalizable element of
which has exactly one eigenvalue not equal to 1. A lattice is a free module. A Weyl group is a finite subgroup of
Let be an index set for the reflections in so that, for
the fixed point space of the transformation The chambers are the connected components of the complement
of these hyperplanes in These are fundamental regions for the action of
Let be a nondegenerate invariant bilinear form on Fix a chamber and choose vectors
The alcoves are the connected components of the complement
in The fundamental alcove is the alcove
where is the closure of An example is the case of type where the picture is
The translation in is the operator
The reflection in the hyperplane is given by
The extended affine Weyl group is
Denote the walls of by
and extend this indexing so that
the fundamental alcove. Then the affine Weyl group,
the reflections in the hyperplanes
Furthermore, is a fundamental region for te action of on and so there is a bijection
The length of
The difference between and is the group
The group is the set of elements of of length 0. An element of acts on the fundamental alcove by an automorphism. Its action on induces a permutation of the walls of and hence a permutation of
If and let be the image of the origin under the action of on If denotes the reflection in the wall of and denotes the longest element of the stabilizer of in then
The group acts freely on
( copies of tiled by alcoves) so that is in the same spot as except on the "sheet" of
It is helpful to think of the elements of as the deck transformations which transfer between the sheets in
is a bijection. In type the two sheets in
where the numbering on the walls of the alcoves is equivariant so that, for the numbering on the walls of is the image of the numbering on the walls of
The 0-polygon is the orbit of in
the polygon is
the translate of the orbit of by The space
is tiled by the polygons and, via (2.11 [Ref?]), we make identifications between
and their geometric counterparts in
so that is a set of representatives of the orbits of the action of on The fundamental weights are the generators
of the module so that
The lattice has basis
and the map
is a bijection. The simple coroots are
the dual basis to the fundamental weights,
The dominance order is the partial order on given by
In type the lattice
an orthonormal basis of
is the dihedral group of order 8 generated by the reflections and in the hyperplanes and respectively, where
In this case
Notes and References
The above notes are taken from section 2 of the paper
Alcove Walks, Hecke Algebras, Spherical Functions, Crystals and Column Strict Tableaux, Pure and Applied Mathematics Quarterly 2 no. 4 (2006), 963-1013.
They are also a rtyping into MathML of the notes at http://researchers.ms.unimelb.edu.au/~aram@unimelb/Notes2005/affWeyl12.14.05.pdf
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