102 Affine Coxeter groups and Hecke algebras

In this chapter we describe functions dealing with affine Coxeter groups and Hecke algebras.

We follow the presentation in Kac, §1.1 and 3.7.

A generalized Cartan matrix C is a matrix of integers

of size n× n and of rank l such that ci,i=2, ci,j ≤ 0 is i ≠ j, and ci,j=0 if and only if cj,i=0. We say that C is indecomposable if it does not admit any block decomposition.

Let C be a generalized Cartan matrix. For I a subset of {1,...,n} we denote by CI the square submatrix with indices i,j taken in I. If v is a real vector of length n, we write v>0 if for all i∈{1,...,n} we have vi>0. It can be shown that C is a Cartan matrix if and only if for all sets I, we have det CI>0; or equivalently, if and only if there exists v>0 such that C.v>0. C is called an affine Cartan matrix if for all proper subsets I we have det CI>0, but det C=0; or equivalently if there exists v>0 such that C.v=0.

Given an irreducible Weyl group W with Cartan matrix C, we can construct a generalized Cartan matrix ~ C as follows. Let α0 be the opposed of the highest root. Then the matrix

(
CC.α0
α0.C2
)
is an affine Cartan matrix. The affine Cartan matrices which can be obtained in this way are those we are interested in, which give rise to affine Weyl groups.

Let d=n-l. A realization of a generalized Cartan matrix is a pair V,V of vector spaces of dimension n+d together with vectors α1,...,αn∈ V (the simple roots), α1,...,αn∈ V (the simple coroots), such that i, αj)=ci,j. Up to isomorphism, a realization is obtained as follows: write C=(
C1
C2
)
where C1 is of rank l. Then take αi to be the first n vectors in a basis of V, and take αj to be given in the dual basis by the rows of the matrix

(
C10
C2Idd
).
To C we associate a reflection group in the space V, generated by the fundamental reflections ri given by ri(v)=v-(αi,v)αi. This is a Coxeter group, called the Affine Weyl group ~ W associated to W when we start with the affine Cartan matrix associated to a Weyl group W.

The Affine Weyl group is infinite; it has one additional generator s0 (the reflection with respect to α0) compared to W. In GAP3 we can not use 0 as a label by default for a generator of a Coxeter group (because the default labels are used as indices, and indices start at 1 in GAP3) so we label it as n+1 where n is the numbers of generators of W. The user can change that by setting the field .reflectionsLabels of ~ W to Concatenation([1..n],[0]). As in the finite case, we associate to the realization of ~ W a Dynkin diagram. We get the following diagrams:

    gap> PrintDiagram(Affine(CoxeterGroup("A",1))); # infinite bond
    A1~  1 oo 2
    gap> PrintDiagram(Affine(CoxeterGroup("A",5)));  # for An, n not 1
            - - - 6 - - -
           /             \ 
    A5~   1 - 2 - 3 - 4 - 5
    gap> PrintDiagram(Affine(CoxeterGroup("B",4)));  # for Bn
                  5
                  |
    B4~   1 < 2 - 3 - 4
    gap> PrintDiagram(Affine(CoxeterGroup("C",4)));  # for Cn
    C4~   1 > 2 - 3 - 4 < 5
    gap> PrintDiagram(Affine(CoxeterGroup("D",6)));  # for Dn
    D6~  1           7
          \         /
           3 - 4 - 5
          /         \ 
         2           6
    gap> PrintDiagram(Affine(CoxeterGroup("E",6)));
                 7
                 |
                 2
                 |
    E6~  1 - 3 - 4 - 5 - 6
    gap> PrintDiagram(Affine(CoxeterGroup("E",7)));
                     2
                     |
    E7~  8 - 1 - 3 - 4 - 5 - 6 - 7
    gap> PrintDiagram(Affine(CoxeterGroup("E",8)));
                 2
                 |
    E8~  1 - 3 - 4 - 5 - 6 - 7 - 8 - 9
    gap> PrintDiagram(Affine(CoxeterGroup("F",4)));
    F4~  5 - 1 - 2 > 3 - 4
    gap> PrintDiagram(Affine(CoxeterGroup("G",2)));
    G2~  3 - 1 > 2

We represent in GAP3 the group ~ W as a matrix group in the space V.

Subsections

  1. Affine
  2. Operations and functions for Affine Weyl groups
  3. AffineRootAction

102.1 Affine

Affine( W )

This function returns the affine Weyl corresponding to the Weyl group W.

102.2 Operations and functions for Affine Weyl groups

All matrix group operations are defined on Affine Weyl groups, as well as all functions defined for abstract Coxeter groups (in particular Hecke algebras and their Kazhdan-Lusztig bases).

The functions Print(W) and PrintDiagram(W) are also defined and print an appropriate representation of W:

    gap> W:=Affine(CoxeterGroup("A",4));
    Affine(CoxeterGroup("A",4))
    gap> PrintDiagram(W);
            - - 5 - -
           /         \ 
    A4~   1 - 2 - 3 - 4

The function ReflectionLength is also defined, using the formula of Lewis, McCammond, Petersen and Schwer.

102.3 AffineRootAction

AffineRootAction(W,w,x)

The Affine Weyl group W can be realized as affine transformations on the vector space spanned by the roots of W.linear. Given a vector x expressed in the basis of simple roots of W.linear and w in W, this function returns returns the image of x under w realized as an affine transformation.

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gap3-jm
19 Feb 2018