75.5 CoxeterGroup

CoxeterGroup( simpleRoots, simpleCoroots[, omega] )

CoxeterGroup( C[, "sc"][, omega] )

CoxeterGroup( type1, n1, ... , typek, nk[, "sc"][, omega] )

CoxeterGroup( rec )

This function returns a permutation group record containing the basic information about the Coxeter group and the root system determined by its arguments. In the first form a set of roots is given explicitly as the lines of the matrix simpleRoots, representing vectors in a vector space V, as well as a set of coroots as the lines of the matrix simpleCoroots expressed in the dual basis of Vdual. The product C=simpleCoroots*TransposedMat(simpleRoots) must be a valid Cartan matrix. The dimension of V can be greater than Length(C). The length of C is called the semisimple rank of the Coxeter datum, while the dimension of V is called its rank.

In the second form C is a Cartan matrix, and the call CoxeterGroup(C) is equivalent to

CoxeterGroup(IdentityMat(Length(C)),C).

In this case, the root system is embedded in the lattice of integral vectors of V like the root system of an adjoint algebraic reductive group in the lattice of characters of a maximal torus.

If the optional "sc" argument is given, the situation is reversed: the simple coroots are given by the identity matrix, and the simple roots by the transposed of C (this corresponds to the embedding of the root system in the lattice of characters of a maximal torus in a simply connected algebraic group).

The third form is equivalent to

CoxeterGroup(CartanMat(type1, n1, ..., typek, nk) [, "sc"][, omega]).

The resulting record, that we will call a Coxeter datum, has additional entries describing various information on the root system and Coxeter group that we describe below.

The argument omega in one of the first three forms can be used to specify a group of automorphisms of the Coxeter datum, that is, a group of invertible linear transformations of V which preserve the set of roots and whose adjoint maps preserve the set of coroots. When the rank is equal to the semisimple rank (we then say that the Coxeter datum is semisimple), this can be given as a permutation group (on the roots). Otherwise it must be given as a matrix group.

The last form takes as an argument a record which has a field coxeter and returns the value of this field. This is used to return the Coxeter group of objects derived from Coxeter groups, such as Coxeter cosets, Hecke algebras and braid elements.

We document the following entries in a Coxeter datum record which are guaranteed to remain present in future versions of the package. Other undocumented entries should not be relied upon, they may change without notice.

isCoxeterGroup, isDomain, isGroup, isPermGroup, isFinite:

true

cartan:

the Cartan matrix C

simpleRoots:

the matrix of simple roots

simpleCoroots:

the matrix of simple coroots

semisimpleRank:

the length of C

rank:

the length of TransposedMat(.simpleRoots)

N:

the number of positive roots

roots:

the root vectors, given as linear combinations of fundamental roots (note that in a former version of the package only the positive roots were stored). The first N roots are positive, the next N are the corresponding negative roots. Moreover, the first semisimpleRank roots are the fundamental roots corresponding to the rows of C. The positive roots are ordered by increasing height.

coroots:

the same information for the coroots. The coroot corresponding to a given root is in the same relative position in the list of coroots as the root in the list of roots.

rootLengths:

the vector of length of roots. The shortest roots in an irreducible subsystem are given the length 1, the others length 2 (or 3 in type G_2).

orbitRepresentative:

this is a list of same length as roots, which for each root, gives the smallest index of a root in the same W-orbit.

orbitRepresentativeElements:

a list of same length as roots, which for the i-th root, gives an element w of W of minimal length such that i=orbitRepresentative[i]^w.

matgens:

the matrices (in row convention) of the simple reflections of the Coxeter group with respect to the basis consisting of the fundamental root vectors.

generators:

the generators as permutations of the root vectors. They are given in the same order as the first semisimpleRank roots.

omega:

the value of the argument omega if it has been specified. Otherwise unbound.

    gap> W := CoxeterGroup( "A", 4 );;
    gap> PrintArray( W.cartan );
    [ [   2,  -1,   0,   0 ],
      [  -1,   2,  -1,   0 ],
      [   0,  -1,   2,  -1 ],
      [   0,   0,  -1,   2 ] ]
    gap> W.matgens;
    [ [ [ -1, 0, 0, 0 ], [ 1, 1, 0, 0 ], [ 0, 0, 1, 0 ], [ 0, 0, 0, 1 ] ],
      [ [ 1, 1, 0, 0 ], [ 0, -1, 0, 0 ], [ 0, 1, 1, 0 ], [ 0, 0, 0, 1 ] ],
      [ [ 1, 0, 0, 0 ], [ 0, 1, 1, 0 ], [ 0, 0, -1, 0 ], [ 0, 0, 1, 1 ] ],
      [ [ 1, 0, 0, 0 ], [ 0, 1, 0, 0 ], [ 0, 0, 1, 1 ], [ 0, 0, 0, -1 ] ]
     ]
    gap> W.roots;
    [ [ 1, 0, 0, 0 ], [ 0, 1, 0, 0 ], [ 0, 0, 1, 0 ], [ 0, 0, 0, 1 ], 
      [ 1, 1, 0, 0 ], [ 0, 1, 1, 0 ], [ 0, 0, 1, 1 ], [ 1, 1, 1, 0 ], 
      [ 0, 1, 1, 1 ], [ 1, 1, 1, 1 ], [ -1, 0, 0, 0 ], [ 0, -1, 0, 0 ], 
      [ 0, 0, -1, 0 ], [ 0, 0, 0, -1 ], [ -1, -1, 0, 0 ], 
      [ 0, -1, -1, 0 ], [ 0, 0, -1, -1 ], [ -1, -1, -1, 0 ], 
      [ 0, -1, -1, -1 ], [ -1, -1, -1, -1 ] ]

This function requires the package "chevie" (see RequirePackage).

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