.. _SectGrpRflGrpRflOp: .. _operations: Operations on Reflection Groups ================================ See Chapter :ref:`ChapGrpMat` for general functions for matrix groups. Note that most of the functions in this section only work for real reflection groups. .. magma:function:: IsCoxeterIsomorphic(W1, W2) :input_types: GrpMat, GrpMat :output_types: BoolElt :label: IsCoxeterIsomorphic_GrpMat_GrpMat Returns ``true`` if and only if the real reflection groups :math:`W_1` and :math:`W_2` are isomorphic as Coxeter systems. .. magma:function:: IsCartanEquivalent(W1, W2) :input_types: GrpMat, GrpMat :output_types: BoolElt :label: IsCartanEquivalent_GrpMat_GrpMat Returns ``true`` if and only if the crystallographic real reflection groups :math:`W_1` and :math:`W_2` have Cartan equivalent Cartan matrices. .. magma:example:: Example: Isomorphism :label: Isomorphism .. code-block:: magma > W1 := ReflectionGroup("B3"); > W2 := ReflectionGroup("C3"); > IsCoxeterIsomorphic(W1, W2); true [ 1, 2, 3 ] > IsCartanEquivalent(W1, W2); false %%a> assert not $1; .. magma:function:: CartanName(W) :input_types: GrpMat :output_types: List :label: CartanName_GrpMat The Cartan name of the finite or affine real reflection group :math:`W` (Section :ref:`SectCartanFinAff`). .. magma:function:: CoxeterDiagram(W) :input_types: GrpMat :label: CoxeterDiagram_GrpMat A display of the Coxeter diagram of the real reflection group :math:`W` (Section :ref:`SectCartanFinAff`). If :math:`W` is not affine or finite, an error is flagged. .. magma:function:: DynkinDiagram(W) :input_types: GrpMat :label: DynkinDiagram_GrpMat A display of the Coxeter diagram of the real reflection group :math:`W` (Section :ref:`SectCartanFinAff`). If :math:`W` is not affine or finite, or if :math:`W` is not crystallographic, an error is flagged. .. magma:example:: Example: Name And Diagram :label: NameAndDiagram .. code-block:: magma > G := CompleteGraph(3); > W := ReflectionGroup(G); > CartanName(W); A~2 > CoxeterDiagram(W); A~2 1 - 2 | | - 3 - .. magma:function:: RootSystem(W) :input_types: GrpMat :output_types: RootDtm :label: RootSystem_GrpMat The root system of the finite real reflection group :math:`W` (Chapter :ref:`ChapRootSys`). If :math:`W` is infinite, an error is flagged. .. magma:function:: RootDatum(W) :input_types: GrpMat :output_types: RootDtm :label: RootDatum_GrpMat The root datum of the finite real reflection group :math:`W` (Chapter :ref:`ChapRootDtm`). The roots and coroots of :math:`W` must have integral components, and :math:`W` must be finite. .. magma:function:: CoxeterMatrix(W) :input_types: GrpMat :output_types: AlgMatElt :label: CoxeterMatrix_GrpMat The Coxeter matrix of the real reflection group :math:`W` (Section :ref:`SectCartanCoxMat`). .. magma:function:: CoxeterGraph(W) :input_types: GrpMat :output_types: GrphUnd :label: CoxeterGraph_GrpMat The Coxeter graph of the real reflection group :math:`W` (Section :ref:`SectCartanCoxGrph`). .. magma:function:: CartanMatrix(W) :input_types: GrpMat :output_types: AlgMatElt :label: CartanMatrix_GrpMat The Cartan matrix of the real reflection group :math:`W` (Section :ref:`SectCartanCarMat`). .. magma:function:: DynkinDigraph(W) :input_types: GrpMat :output_types: GrphDir :label: DynkinDigraph_GrpMat The Dynkin digraph of the real reflection group :math:`W` (Section :ref:`SectCartanDynDigrph`). .. magma:function:: Rank(W) :input_types: GrpMat :output_types: RngIntElt :label: Rank_GrpMat .. magma:function:: NumberOfGenerators(W) :input_types: GrpMat :output_types: RngIntElt :label: NumberOfGenerators_GrpMat The rank of the reflection group :math:`W`. .. magma:example:: Example: Rank Dimension :label: RankDimension .. code-block:: magma > R := StandardRootSystem("A", 4); > W := ReflectionGroup(R); > Rank(W); 4 %%a> assert $1 eq 4; > Dimension(W); 5 %%a> assert $1 eq 5; .. magma:function:: FundamentalGroup(W) :input_types: GrpMat :output_types: GrpAb :label: FundamentalGroup_GrpMat The fundamental group of the real reflection group :math:`W` (Subsection :ref:`SubsectRDIsogeny`). The roots and coroots of :math:`W` must have integral components. .. magma:function:: IsogenyGroup(W) :input_types: GrpMat :output_types: GrpAb, Map :label: IsogenyGroup_GrpMat The isogeny group of the real reflection group :math:`W`, together with the injection into the fundamental group (Subsection :ref:`SubsectRDIsogeny`). The roots and coroots of :math:`W` must have integral components. .. magma:function:: CoisogenyGroup(W) :input_types: GrpMat :output_types: GrpAb, Map :label: CoisogenyGroup_GrpMat The fundamental group of the real reflection group :math:`W` together with the projection onto the fundamental group (Subsection :ref:`SubsectRDIsogeny`). The roots and coroots of :math:`W` must have integral components. .. magma:function:: BasicDegrees(W) :input_types: GrpMat :output_types: RngIntElt :label: BasicDegrees_GrpMat The degrees of the basic invariant polynomials of the reflection group :math:`W`. These are computed using the table in :raw-latex:`\cite[page 155]{Carter-small}` if the group is real, and using the algorithm of :cite:`LehrerTaylor` in other cases. If :math:`W` is infinite, an error is flagged. .. magma:function:: BasicCodegrees(W) :input_types: GrpMat :output_types: RngIntElt :label: BasicCodegrees_GrpMat The basic codegrees of the reflection group :math:`W`. These are computed using the algorithm of :cite:`LehrerTaylor`. If :math:`W` is infinite, an error is flagged. .. magma:example:: Example: Basic Degrees :label: BasicDegrees The product of the basic degrees is the order of the Coxeter group; the sum of the basic degrees is the sum of the rank and the number of positive roots. .. code-block:: magma > W := ReflectionGroup("E6"); > degs := BasicDegrees(W); > degs; [ 2, 5, 6, 8, 9, 12 ] > &*degs eq #W; true %%a> assert $1; > &+degs eq NumPosRoots(W) + Rank(W); true %%a> assert $1; .. magma:function:: LongestElement(W) :input_types: GrpMat :output_types: SeqEnum :label: LongestElement_GrpMat The unique longest element in the finite real reflection group :math:`W`. .. magma:function:: CoxeterElement(W) :input_types: GrpMat :output_types: SeqEnum :label: CoxeterElement_GrpMat The Coxeter element in the reflection group :math:`W`, ie. the product of the generators. .. magma:function:: CoxeterNumber(W) :input_types: GrpMat :output_types: SeqEnum :label: CoxeterNumber_GrpMat The order of the Coxeter element in the real reflection group :math:`W`. .. magma:example:: Example: Operations :label: Operations Operations on groups. .. code-block:: magma > W := ReflectionGroup("A4"); > LongestElement(W); [ 0 0 0 -1] [ 0 0 -1 0] [ 0 -1 0 0] [-1 0 0 0] > CoxeterElement(W); [-1 -1 -1 -1] [ 1 0 0 0] [ 0 1 0 0] [ 0 0 1 0] .. magma:function:: LeftDescentSet(W, w) :input_types: GrpMat, GrpMatElt :output_types: $\{\}$ :label: LeftDescentSet_GrpMat_GrpMatElt The set of indices :math:`r` of simple roots of the finite real reflection group :math:`W` such that the length of the product :math:`s_rw` is less than that of the element :math:`w`. .. magma:function:: RightDescentSet(W, w) :input_types: GrpMat, GrpMatElt :output_types: $\{\}$ :label: RightDescentSet_GrpMat_GrpMatElt The set of indices :math:`r` of simple roots of the finite real reflection group :math:`W` such that the length of the product :math:`ws_r` is less than that of the element :math:`w`. .. magma:example:: Example: Descent Sets :label: DescentSets .. code-block:: magma > W := ReflectionGroup("A5"); > x := W.1*W.2*W.4*W.5; > LeftDescentSet(W, x); { 1, 4 } > RightDescentSet(W, x); { 2, 5 }