CoxeterCoset( W[, F0Mat ] )
CoxeterCoset( W[, F0Perm] )
This function returns a Coxeter coset as a GAP object. The argument
W must be a Coxeter group (created by CoxeterGroup
or
ReflectionSubgroup
). In the first form the argument F0Mat must be an
invertible matrix with Rank(W)
rows, representing an automorphism
F_0 of the root system of the parent of W. In the second form
F0Perm is a permutation which describes the images of the simple roots
under F_0 (and only these images are used). Of course this form is
only allowed if the semisimple rank of W equals the rank (i.e., the
simple roots are a basis of V). If there is no second argument the
default for F0Mat is the identity matrix.
CoxeterCoset
returns a record from which we document the following
components:
isDomain
, isFinite
:
coxeter
:
F0Mat
:
F0Perm
:F0Mat
phi
:
w1
:phi
/F0Perm
In the first example we create a Coxeter coset corresponding to the general unitary groups GU_3(q) over finite fields with q elements.
gap> W := CoxeterGroup( [ [ 1, -1, 0 ], [ 0, 1, -1 ] ], > [ [ 1, -1, 0 ], [ 0, 1, -1 ] ] );; gap> gu3 := CoxeterCoset( W, -IdentityMat( 3 ) ); CoxeterCoset(CoxeterGroup([ [ 1, -1, 0 ], [ 0, 1, -1 ] ], [ [ 1, -1, 0 ], [ 0, 1, -1 ] ]), [ [ -1, 0, 0 ], [ 0, -1, 0 ], [ 0, 0, -1 ] ]) gap> F4 := CoxeterGroup( "F", 4 );; gap> D4 := ReflectionSubgroup( F4, [ 1, 2, 16, 48 ] ); ReflectionSubgroup(CoxeterGroup("F", 4), [ 1, 2, 9, 16 ]) gap> PrintDynkinDiagram( D4 ); D4 9 \ 1 - 16 / 2 gap> 3D4 := CoxeterCoset( D4, (2,9,16) ); CoxeterCoset(ReflectionSubgroup(CoxeterGroup("F", 4), [ 1, 2, 9, 16 ]), ( 2, 9,16))
These functions require the package "chevie" (see RequirePackage).
GAP 3.4.4