An element of a Coxeter group is called a reflection if it is conjugate to one of the Coxeter generators. A reflection subgroup of a Coxeter group is a subgroup which is generated by a set of reflections---reflection subgroups are also Coxeter groups. The most important class of reflection subgroups are the standard parabolic subgroups, which are generated by a subset of the simple roots. Note that in a reflection subgroup, the elements are given as permutations of the roots of the larger group.
The functions in this section are currently only implemented for groups with a root datum.
The reflection subgroup of the Coxeter group W generated by the roots alpha_(a_1), ..., alpha_(a_k) where a={a_1, ..., a_k} is a set of integers.
The reflection subgroup of the Coxeter group W generated by simple roots alpha_(s_1), ..., alpha_(s_k) where s=[s_1, ..., s_k] is a sequence of integers. In this version the roots must be simple in the root subdatum (ie. none of them may be a summand of another) otherwise an error is signalled. The simple roots will appear in the reflection subgroup in the given order.
The standard parabolic subgroup of the Coxeter group W generated by the simple roots alpha_(a_1), ..., alpha_(a_k) where a={a_1, ..., a_k}subseteq{1, ..., Rank(W)}.
Returns true if, and only if, H is a reflection subgroup of the Coxeter group W.
Returns true if, and only if, H is a standard parabolic subgroup of the Coxeter group W.
The overgroup of H, ie. the Coxeter group whose roots are permuted by the elements of the Coxeter subgroup H.
The root datum whose roots are permuted by the elements of the Coxeter subgroup H.
Given a Coxeter subgroup H this returns the Coxeter group L isomorphic to H but acting on the roots of H itself rather than the roots of its overgroup, together with the isomorphism L to H.
> W := CoxeterGroup( "A4" );
> P := StandardParabolicSubgroup( W, {1,2} );
> Overgroup( P ) eq W;
true
> L, h := LocalCoxeterGroup( P );
> hinv := Inverse( h );
> L.1;
(1, 4)(2, 3)(5, 6)
> h(L.1);
(1, 11)(2, 5)(6, 8)(9, 10)(12, 15)(16, 18)(19, 20)
> hinv(h(L.1));
(1, 4)(2, 3)(5, 6)
The indexed set of (right) coset representatives of the reflection subgroup H of the Coxeter group W. This contains the unique element of shortest length in each coset.
The representative of the coset Hx in the Coxeter group W. This is the unique element of Hx of shortest length in W and also the unique element of Hx which sends every positive root of H to another positive root.
> W := CoxeterGroup( "A4" );
> P := StandardParabolicSubgroup( W, {1,2} );
> x := W.1 * W.2 * W.3;
> x := TransversalElt( W, P, x );
> x eq W.3;
true
> x in Transversal( W, P );
true
The direct sum of the Coxeter groups W_1 and W_2.
The dual of the root datum W, obtained by swapping the roots and coroots.
> W1 := CoxeterGroup( "G2" );
> W2 := CoxeterGroup( "C3" );
> W1 + Dual(W2);
Coxeter group: Permutation group acting on a set of cardinality 30
Order = 576 = 2^6 * 3^2
(1, 7)(2, 5)(3, 4)(8, 11)(9, 10)
(1, 3)(2, 8)(5, 6)(7, 9)(11, 12)
(13, 22)(14, 16)(17, 20)(19, 21)(23, 25)(26, 29)(28, 30)
(13, 16)(14, 23)(15, 17)(18, 21)(22, 25)(24, 26)(27, 30)
(14, 19)(15, 24)(16, 21)(23, 28)(25, 30)