.. _SectGrpFPCoxAction: .. _action: Roots, Coroots and Reflections ============================== Many of these functions have an optional argument ``Basis`` which may take one of the following values 1. ``"Standard"``: the standard basis for the (co)root space. This is the default. 2. ``"Root"``: the basis of simple (co)roots. 3. ``"Weight"``: the basis of fundamental (co)weights (see Subsection :ref:`SubsectRDRootWeight` below). .. _SubsectRDRootAccess: .. _access: Accessing Roots and Coroots --------------------------- .. magma:function:: RootSpace(W) :input_types: GrpMat :output_types: Lat :label: RootSpace_GrpMat .. magma:function:: CorootSpace(W) :input_types: GrpMat :output_types: Lat :label: CorootSpace_GrpMat The base space of the reflection group :math:`W`. If :math:`W` is not a reflection group, an error occurs. .. magma:example:: Example: Root Space :label: RootSpace .. code-block:: magma > W := ComplexReflectionGroup("M", 3); > RootSpace(W); Full Vector space of degree 3 over Cyclotomic Field of order 24 and degree 8 .. magma:function:: SimpleOrders(W) :input_types: GrpMat :output_types: [RngIntElt] :label: SimpleOrders_GrpMat The sequence of simple orders of the reflection group :math:`W`. If :math:`W` is not a reflection group, an error is flagged. .. magma:function:: SimpleRoots(W) :input_types: GrpMat :output_types: Mtrx :label: SimpleRoots_GrpMat .. magma:function:: SimpleCoroots(W) :input_types: GrpMat :output_types: Mtrx :label: SimpleCoroots_GrpMat The simple (co)roots of the reflection group :math:`W` as the rows of a matrix, i.e. :math:`A` (resp. :math:`B`). .. magma:function:: NumberOfPositiveRoots(W) :input_types: GrpMat :output_types: RngIntElt :label: NumberOfPositiveRoots_GrpMat .. magma:function:: NumPosRoots(W) :input_types: GrpMat :output_types: RngIntElt :label: NumPosRoots_GrpMat The number of positive roots of the real reflection group :math:`W`. This is also the number of positive coroots. The total number of (co)roots is twice the number of positive (co)roots. This number is finite if and only if :math:`W` is finite. .. magma:function:: Roots(W) :input_types: GrpMat :output_types: $\{@@\}$ :label: Roots_GrpMat .. magma:function:: Coroots(W) :input_types: GrpMat :output_types: $\{@@\}$ :label: Coroots_GrpMat :parameters: Basis : MonStgElt : ``Standard" The indexed set of (co)roots of the real reflection group :math:`W`, i.e. :math:`\{@\,\alpha_1,\dots\alpha_{2N}\,@\}` (resp. :math:`\{@\,\alpha_1^\star,\dots\alpha_{2N}^\star\,@\}`). If :math:`W` is infinite, an error is flagged. .. magma:function:: PositiveRoots(W) :input_types: GrpMat :output_types: $\{@@\}$ :label: PositiveRoots_GrpMat .. magma:function:: PositiveCoroots(W) :input_types: GrpMat :output_types: $\{@@\}$ :label: PositiveCoroots_GrpMat :parameters: Basis : MonStgElt : ``Standard" The indexed set of positive (co)roots of the real reflection group :math:`W`, that is, :math:`\{@\,\alpha_1,\dots\alpha_N\,@\}` (resp. :math:`\{@\,\alpha_1^\star,\dots\alpha_N^\star\,@\}`). If :math:`W` is infinite, an error is flagged. .. magma:function:: Root(W, r) :input_types: GrpMat, RngIntElt :output_types: $\{@@\}$ :label: Root_GrpMat_RngIntElt .. magma:function:: Coroot(W, r) :input_types: GrpMat, RngIntElt :output_types: $\{@@\}$ :label: Coroot_GrpMat_RngIntElt :parameters: Basis : MonStgElt : ``Standard" The :math:`r`\ th (co)root :math:`\alpha_r` (resp. :math:`\alpha_r^\star`) of the real reflection group :math:`W`. If :math:`W` is infinite, an error is flagged. .. magma:function:: RootPosition(W, v) :input_types: GrpMat, . :output_types: $\{@@\}$ :label: RootPosition_GrpMat .. magma:function:: CorootPosition(W, v) :input_types: GrpMat, . :output_types: $\{@@\}$ :label: CorootPosition_GrpMat :parameters: Basis : MonStgElt : ``Standard" If :math:`v` is a (co)root in the finite real reflection group :math:`W`, return its index; otherwise return 0. These functions will try to coerce :math:`v` into the appropriate lattice; :math:`v` should be written with respect to the basis specified by the parameter ``Basis``. If :math:`W` is infinite, an error is flagged. .. magma:example:: Example: Roots Coroots :label: RootsCoroots .. code-block:: magma > W := ReflectionGroup("A3"); > Roots(W); {@ (1 0 0), (0 1 0), (0 0 1), (1 1 0), (0 1 1), (1 1 1), (-1 0 0), (0 -1 0), (0 0 -1), (-1 -1 0), (0 -1 -1), (-1 -1 -1) @} > PositiveCoroots(W); {@ (2 -1 0), (-1 2 -1), (0 -1 2), (1 1 -1), (-1 1 1), (1 0 1) @} > #Roots(W) eq 2*NumPosRoots(W); true %%a> assert $1; > Root(W, 4); (1 1 0) > Root(W, 4 : Basis := "Root"); (1 1 0) > RootPosition(W, [1,1,0]); 4 %%a> assert $1 eq 4; > A := Matrix(3,3,[1,0,0, -1,-1,-3, 1,2,4]); > B := Matrix(3,3,[2,-1,0, -1,2,-1, 0,1,0]); > W := ReflectionGroup(A,B); > Roots(W); {@ (1 0 0), (-1 -1 -3), (1 2 4), (0 -1 -3), (0 1 1), (1 1 1), (-1 0 0), (1 1 3), (-1 -2 -4), (0 1 3), (0 -1 -1), (-1 -1 -1) @} > PositiveCoroots(W); {@ (2 -1 0), (-1 2 -1), (0 1 0), (1 1 -1), (-1 3 -1), (1 2 -1) @} > #Roots(W) eq 2*NumPosRoots(W); true %%a> assert $1; > Root(W, 4); (0 -1 -3) > Root(W, 4 : Basis := "Root"); (1 1 0) > RootPosition(W, [0,-1,-3]); 4 %%a> assert $1 eq 4; .. _SubsectRDRootAction: .. _rootrefl: Reflections ----------- The root :math:`\alpha` acts on the root space via the reflection :math:`s_\alpha`; the coroot :math:`\alpha^\star` acts on the coroot space via the coreflection :math:`s_\alpha^\star`. .. magma:function:: ReflectionMatrices(W) :input_types: GrpMat :output_types: [AlgMatElt] :label: ReflectionMatrices_GrpMat .. magma:function:: CoreflectionMatrices(W) :input_types: GrpMat :output_types: [AlgMatElt] :label: CoreflectionMatrices_GrpMat Basis : MonStgElt : “Standard" The sequence of reflections in the finite real reflection group :math:`W`. The :math:`r`\ th reflection in the sequence corresponds to the :math:`r`\ th (co)root. .. magma:function:: SimpleReflectionMatrices(W) :input_types: GrpMat :output_types: [AlgMatElt] :label: SimpleReflectionMatrices_GrpMat .. magma:function:: SimpleCoreflectionMatrices(W) :input_types: GrpMat :output_types: [AlgMatElt] :label: SimpleCoreflectionMatrices_GrpMat Basis : MonStgElt : “Standard" The matrices giving the action of the simple (co)roots on the (co)root space of the finite real reflection group :math:`W`. .. magma:function:: ReflectionMatrix(W, r) :input_types: GrpMat, RngIntElt :output_types: AlgMatElt :label: ReflectionMatrix_GrpMat_RngIntElt .. magma:function:: CoreflectionMatrix(W, r) :input_types: GrpMat, RngIntElt :output_types: [] :label: CoreflectionMatrix_GrpMat_RngIntElt Basis : MonStgElt : “Standard" The reflection in finite real reflection group :math:`W` corresponding to the :math:`r`\ th (co)root. If :math:`r=1,\dots,n`, this is a generator of :math:`W`. .. magma:function:: SimpleReflectionPermutations(W) :input_types: GrpMat :output_types: [] :label: SimpleReflectionPermutations_GrpMat The sequence of permutations giving the action of the simple (co)roots of the finite reflection group :math:`W` on the (co)roots. This action is the same for roots and coroots. .. magma:function:: ReflectionPermutations(W) :input_types: GrpMat :output_types: [] :label: ReflectionPermutations_GrpMat The sequence of permutations giving the action of the (co)roots of the finite reflection group :math:`W` on the (co)roots. This action is the same for roots and coroots. .. magma:function:: ReflectionPermutation(W, r) :input_types: GrpMat, RngIntElt :output_types: [] :label: ReflectionPermutation_GrpMat_RngIntElt The permutation giving the action of the :math:`r`\ th (co)root of the finite reflection group :math:`W` on the (co)roots. This action is the same for roots and coroots. .. magma:function:: ReflectionWords(W) :input_types: GrpMat :output_types: [] :label: ReflectionWords_GrpMat The sequence of words in the simple reflections for all the reflections of the real reflection group :math:`W`. These words are given as sequences of integers. In other words, if :math:`a = [a_1,\dots,a_l] = \hbox{\tt ReflectionWords(W)[r]}`, then :math:`s_{\alpha_r} = s_{\alpha_{a_1}} \cdots s_{\alpha_{a_l}}`. .. magma:function:: ReflectionWord(W, r) :input_types: GrpMat, RngIntElt :output_types: [] :label: ReflectionWord_GrpMat_RngIntElt The word in the simple reflections for the :math:`r`\ th reflection of the real reflection group :math:`W`. The word is given as a sequence of integers. In other words, if :math:`a = [a_1,\dots,a_l] = \hbox{\tt ReflectionWord(W,r)}`, then :math:`s_{\alpha_r} = s_{\alpha_{a_1}} \cdots s_{\alpha_{a_l}}`. .. magma:example:: Example: Action :label: Action .. code-block:: magma > Q := RationalField(); > W := ReflectionGroup("A3"); > mx := ReflectionMatrix(W, 4); > perm := ReflectionPermutation(W, 4); > RootPosition(W, Vector(Q, Eltseq(Root(W,2))) * mx) eq 2^perm; true %%a> assert $1; > mx := CoreflectionMatrix(W, 4); > CorootPosition(W, Coroot(W,2) * mx) eq 2^perm; true %%a> assert $1; .. magma:function:: Length(w) :input_types: GrpMatElt :output_types: RngIntElt :label: Length_GrpMatElt .. magma:function:: CoxeterLength(w) :input_types: GrpMatElt :output_types: RngIntElt :label: CoxeterLength_GrpMatElt The length of :math:`w` as an element of the Coxeter group :math:`W`, i.e. the number of positive roots of :math:`W` which become negative under the action of :math:`w`. .. _SubsectRDRootWeight: .. _weights: Weights ------- .. magma:function:: WeightLattice(W) :input_types: GrpMat :output_types: Lat :label: WeightLattice_GrpMat .. magma:function:: CoweightLattice(W) :input_types: GrpMat :output_types: Lat :label: CoweightLattice_GrpMat The (co)weight lattice of the real reflection group :math:`W`. The roots and coroots of :math:`W` must have integral components. .. magma:function:: FundamentalWeights(W) :input_types: GrpMat :output_types: Mtrx :label: FundamentalWeights_GrpMat .. magma:function:: FundamentalCoweights(W) :input_types: GrpMat :output_types: Mtrx :label: FundamentalCoweights_GrpMat :parameters: Basis : MonStgElt : ``Standard" The fundamental weights of the real reflection group :math:`W` given as the rows of a matrix. The roots and coroots of :math:`W` must have integral components. .. magma:example:: Example: Weights :label: Weights .. code-block:: magma > W := ReflectionGroup("E6"); > WeightLattice(W); Lattice of rank 6 and degree 6 Basis: (4 3 5 6 4 2) (3 6 6 9 6 3) (5 6 10 12 8 4) (6 9 12 18 12 6) (4 6 8 12 10 5) (2 3 4 6 5 4) Basis Denominator: 3 > FundamentalWeights(W); [ 4/3 1 5/3 2 4/3 2/3] [ 1 2 2 3 2 1] [ 5/3 2 10/3 4 8/3 4/3] [ 2 3 4 6 4 2] [ 4/3 2 8/3 4 10/3 5/3] [ 2/3 1 4/3 2 5/3 4/3] .. magma:function:: IsDominant(R, v) :input_types: RootDtm, . :output_types: ModTupFldElt, GrpFPCoxElt :label: IsDominant_RootDtm :parameters: Basis : MonStgElt : ``Standard" Returns ``true`` if and only if :math:`v` is a dominant weight for the root datum :math:`R`, ie, a nonnegative integral linear combination of the fundamental weights. .. magma:function:: DominantWeight(W, v) :input_types: GrpMat, . :output_types: ModTupFldElt, GrpFPCoxElt :label: DominantWeight_GrpMat :parameters: Basis : MonStgElt : ``Standard" The unique dominant weight in the same :math:`W`-orbit as :math:`v`, where :math:`W` is a real reflection group and :math:`v` is a weight given as a vector or a sequence representing a vector. The second value returned is a Coxeter group element taking :math:`v` to the dominant weight. .. magma:function:: WeightOrbit(W, v) :input_types: GrpMat, . :output_types: {@ ModTupFldElt @}, [GrpFPCoxElt] :label: WeightOrbit_GrpMat :parameters: Basis : MonStgElt : ``Standard" The :math:`W`-orbit of :math:`v` as an indexed set, where :math:`W` is a real reflection group and :math:`v` is a weight given as a vector or a sequence representing a vector. The first element in the orbit is always dominant. The second value returned is a sequence of Coxeter group words taking the weight :math:`v` to the corresponding element of the orbit. .. magma:example:: Example: Dominant Weights :label: DominantWeights .. code-block:: magma > W := CoxeterGroup("B3"); > DominantWeight(W, [1,-1,0] : Basis:="Weight"); (1 0 0) \$.2 * \$.3 * \$.2 * \$.1 > #WeightOrbit(W, [1,-1,0] : Basis:="Weight"); 6 %%a> assert $1 eq 6;