84.1 CoxeterCoset

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:

true

coxeter:

the Coxeter group W

F0Mat:

the matrix acting on V which represents F_0

F0Perm:

the permutation on the roots of W induced by F0Mat

phi:

the shortest element in the coset (as permutation)

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).

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