.. _SectGrpPermCoxNew: .. _subgroups: Reflection Subgroups ==================== A *reflection subgroup* of a Coxeter group is a subgroup which is generated by a set of reflections. Note that 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. Given a set of indices :math:`J\subseteq \{1,\dots,\hbox{\tt Rank(W)}\}`, the corresponding standard parabolic is denoted :math:`W_J`. A *parabolic subgroup* is a subgroup which is conjugate to a standard parabolic subgroup. Note that in a reflection subgroup, the elements are given as permutations of the roots of the *larger* group. Most of the functions in this section are currently only implemented for permutation Coxeter groups with a root *datum* (rather than a root system). .. magma:function:: ReflectionSubgroup(W, a) :input_types: GrpPermCox, $\{\}$ :output_types: GrpPermCox :label: ReflectionSubgroup_GrpPermCox The reflection subgroup of the permutation Coxeter group :math:`W` generated by the roots :math:`\alpha_{a_1},\dots,\alpha_{a_k}` where :math:`a=\{a_1,\dots,a_k\}` is a set of integers. This only works if :math:`W` has an underlying root datum. .. magma:function:: ReflectionSubgroup(W, s) :input_types: GrpPermCox, [] :output_types: GrpPermCox :label: ReflectionSubgroup_GrpPermCox_2 The reflection subgroup of the permutation Coxeter group :math:`W` generated by simple roots :math:`\alpha_{s_1},\dots,\alpha_{s_k}` where :math:`s=[s_1,\dots,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. This only works if :math:`W` has an underlying root datum. .. magma:function:: StandardParabolicSubgroup(W, J) :input_types: GrpPermCox, $\{\}$ :output_types: GrpPermCox :label: StandardParabolicSubgroup_GrpPermCox The standard parabolic subgroup of the Coxeter group :math:`W` generated by the simple roots :math:`\alpha_{j_1},\dots,\alpha_{j_k}` where :math:`J=\{j_1,\dots,j_k\}\subseteq\{1,\dots,\hbox{\tt Rank(W)}\}`. This function works for both finitely presented and permutation Coxeter groups. .. magma:function:: IsReflectionSubgroup(W, H) :input_types: GrpPermCox, GrpPermCox :output_types: BoolElt :label: IsReflectionSubgroup_GrpPermCox_GrpPermCox Returns ``true`` if, and only if, :math:`H` is a reflection subgroup of the permutation Coxeter group :math:`W`. .. magma:function:: IsParabolicSubgroup(W, H) :input_types: GrpPermCox, GrpPermCox :output_types: BoolElt :label: IsParabolicSubgroup_GrpPermCox_GrpPermCox Returns ``true`` if, and only if, :math:`H` is a parabolic subgroup of the permutation Coxeter group :math:`W`. .. magma:function:: IsStandardParabolicSubgroup(W, H) :input_types: GrpPermCox, GrpPermCox :output_types: BoolElt :label: IsStandardParabolicSubgroup_GrpPermCox_GrpPermCox Returns ``true`` if, and only if, :math:`H` is a standard parabolic subgroup of the permutation Coxeter group :math:`W`. .. magma:function:: Overgroup(H) :input_types: GrpPermCox :output_types: GrpPermCox :label: Overgroup_GrpPermCox The overgroup of :math:`H`, ie. the Coxeter group whose roots are permuted by the elements of the permutation Coxeter subgroup :math:`H`. .. magma:function:: Overdatum(H) :input_types: GrpPermCox :output_types: RootDtm :label: Overdatum_GrpPermCox The root datum whose roots are permuted by the elements of the permutation Coxeter subgroup :math:`H`. .. magma:function:: LocalCoxeterGroup(H) :input_types: GrpPermCox :output_types: GrpPermCox, Map :label: LocalCoxeterGroup_GrpPermCox Given a Coxeter subgroup :math:`H` this returns the Coxeter group :math:`L` isomorphic to :math:`H` but acting on the roots of :math:`H` itself rather than the roots of its overgroup, together with the isomorphism :math:`L\to H`. .. magma:example:: Example: Reflection Subgroups :label: ReflectionSubgroups .. code-block:: magma > W := CoxeterGroup("A4"); > P := StandardParabolicSubgroup(W, {1,2}); > Overgroup(P) eq W; true %%a> assert $1; > 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) .. magma:function:: Transversal(W, H) :input_types: GrpPermCox, GrpPermCox :output_types: {@ @} :label: Transversal_GrpPermCox_GrpPermCox The indexed set of (right) coset representatives of the reflection subgroup :math:`H` of the Coxeter group :math:`W`. This contains the unique element of shortest length in each coset. The algorithm is due to Don Taylor (personal communication). .. magma:function:: TransversalWords(W, H) :input_types: GrpPermCox, GrpPermCox :output_types: {@ @} :label: TransversalWords_GrpPermCox_GrpPermCox The indexed set of words of (right) coset representatives of the reflection subgroup :math:`H` of the Coxeter group :math:`W`. The algorithm is due to Don Taylor (personal communication). .. magma:function:: TransversalElt(W, H, x) :input_types: GrpPermCox, GrpPermCox, GrpPermElt :output_types: GrpPermElt :label: TransversalElt_GrpPermCox_GrpPermCox_GrpPermElt The representative of the coset :math:`Hx` in the Coxeter group :math:`W`. This is the unique element of :math:`Hx` of shortest length in :math:`W` and also the unique element of :math:`Hx` which sends every positive root of :math:`H` to another positive root. The algorithm is due to Don Taylor (personal communication). .. magma:example:: Example: Transversals :label: Transversals .. code-block:: magma > 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 %%a> assert $1; > x in Transversal(W, P); true %%a> assert $1; .. magma:function:: TransversalElt(W, x, H) :input_types: GrpPermCox, GrpPermElt, GrpPermCox :output_types: GrpPermElt :label: TransversalElt_GrpPermCox_GrpPermElt_GrpPermCox The representative of the coset :math:`xH` in the Coxeter group :math:`W`. This is the unique element of :math:`xH` of shortest length in :math:`W` and also the unique element of :math:`xH` which sends every positive root of :math:`H` to another positive root. .. magma:function:: TransversalElt(W, H, x, J) :input_types: GrpPermCox, GrpPermCox, GrpPermElt, GrpPermCox :output_types: GrpPermElt :label: TransversalElt_GrpPermCox_GrpPermCox_GrpPermElt_GrpPermCox The representative of the coset :math:`HxJ` in the Coxeter group :math:`W`. This is the unique element of :math:`HxJ` of shortest length in :math:`W` and also the unique element of :math:`HxJ` which sends every positive root of :math:`HJ` to another positive root. .. magma:function:: Transversal(W, J) :input_types: GrpFPCox, $\{$RngIntElt$\}$ :output_types: $\{@$ GrpFPCoxElt $@\}$ :label: Transversal_GrpFPCox_RngIntElt .. magma:function:: Transversal(W, J, L) :input_types: GrpFPCox, $\{$RngIntElt$\}$, RngIntElt :output_types: $\{@$ GrpFPCoxElt $@\}$ :label: Transversal_GrpFPCox_RngIntElt_RngIntElt The set of right coset representatives of minimal length for the standard parabolic subgroup :math:`W_J \le W`. In the first form :math:`W` must be finite and the result is a full transversal. In the second form :math:`W` may be infinite, but the transversal produced is limited to words of length at most :math:`L`. .. magma:function:: Transversal(W, J, K) :input_types: GrpFPCox, $\{$RngIntElt$\}$, $\{$RngIntElt$\}$ :output_types: [ GrpFPCoxElt ], [ ] :label: Transversal_GrpFPCox_RngIntElt_RngIntElt_2 The sequence of :math:`W_J,W_K`-double cosets representatives of minimal length in :math:`W`. Restricted to :math:`W` finite. The second return value gives the generators of the standard parabolic subgroup :math:`W_J\cap W_K^d` for each double coset representative :math:`d`. .. magma:function:: DirectProduct(W1, W2) :input_types: GrpPermCox, GrpPermCox :output_types: GrpPermCox :label: DirectProduct_GrpPermCox_GrpPermCox The direct product of the Coxeter groups :math:`W_1` and :math:`W_2`. .. magma:function:: Dual(W) :input_types: GrpPermCox :output_types: GrpPermCox :label: Dual_GrpPermCox The dual of the Coxeter group :math:`W`, obtained by swapping the roots and coroots. .. magma:example:: Example: Sum Dual :label: SumDual .. code-block:: magma > W1 := CoxeterGroup("G2"); > W2 := CoxeterGroup("C3"); > DirectProduct(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) > W1 := CoxeterGroup(GrpFPCox, "G2"); > W2 := CoxeterGroup(GrpFPCox, "A2"); > DirectProduct(W1, W2); Coxeter group: Finitely presented group on 4 generators Relations $.1 * $.2 * $.1 = $.2 * $.1 * $.2 $.1 * $.3 = $.3 * $.1 $.1 * $.4 = $.4 * $.1 $.2 * $.3 = $.3 * $.2 $.2 * $.4 = $.4 * $.2 $.3 * $.4 * $.3 = $.4 * $.3 * $.4 $.1^2 = Id($) $.2^2 = Id($) $.3^2 = Id($) $.4^2 = Id($)