.. _SectRDOp: .. _operators-root-dtm: Operations on Root Data ======================= .. magma:operation:: R1 eq R2 :input_types: RootDtm, RootDtm :output_types: BoolElt :label: op_eq_RootDtm_RootDtm Returns ``true`` if, and only if, :math:`R_1` and :math:`R_2` are identical root data. .. magma:function:: IsIsomorphic(R1, R2) :input_types: RootDtm, RootDtm :output_types: BoolElt, [RngIntElt], Map :label: IsIsomorphic_RootDtm_RootDtm Returns ``true`` if, and only if, :math:`R_1` and :math:`R_2` are isomorphic root data. If ``true``, the second value returned is a sequence giving the simple root of :math:`R_2` corresponding to each simple root of :math:`R_1`, and the third value returned is an isomorphism :math:`R_1\to R_2`. This function is currently only implemented for semisimple root data. .. magma:function:: IsCartanEquivalent(R1, R2) :input_types: RootDtm, RootDtm :output_types: BoolElt, SeqEnum :label: IsCartanEquivalent_RootDtm_RootDtm Returns ``true`` if and only if the root data :math:`R_1` and :math:`R_2` are Cartan equivalent; i.e. they have isomorphic Dynkin diagrams. If ``true``, the second value returned is a sequence giving the simple root of :math:`R_2` corresponding to each simple root of :math:`R_1`. .. magma:function:: IsIsogenous(R1, R2) :input_types: RootDtm, RootDtm :output_types: BoolElt, SeqEnum, RootDtm, Map, Map, RootDtm, Map, Map :label: IsIsogenous_RootDtm_RootDtm Returns ``true`` if, and only if, :math:`R_1` and :math:`R_2` are isogenous root data. If ``true``, the subsequent values returned are: a sequence giving the root of :math:`R_2` corresponding to each root of :math:`R_1`, the corresponding adjoint root datum :math:`R_{ad}`, the morphisms :math:`R_{ad}\to R_1` and :math:`R_{ad}\to R_2`, the corresponding simply connected root datum :math:`R_{sc}`, and the morphisms :math:`R_1\to R_{sc}` and :math:`R_2\to R_{sc}`. .. magma:example:: Example: Isomorphism Isogeny :label: IsomorphismIsogeny An example of isogenous root data: An example of distinct isomorphic root data: .. code-block:: magma > R1 := RootDatum("A3"); > R2 := RootDatum("A3" : Isogeny := "SC"); > R1 eq R2; false %%a> assert not $1; > IsIsomorphic(R1, R2); false %%a> assert not $1; > IsCartanEquivalent(R1, R2); true [ 1, 2, 3 ] > IsIsogenous(R1, R2); true [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 ] Adjoint root datum of type A3 Mapping from: RootDtm: ad to RootDtm: ad Mapping from: RootDtm: ad to RootDtm: sc Simply connected root datum of type A3 Mapping from: RootDtm: ad to RootDtm: sc Mapping from: RootDtm: sc to RootDtm: sc > C := CartanMatrix("B2"); > R1 := RootDatum(C); > R2 := RootDatum(Transpose(C)); > R1; R2; Adjoint root datum of type B2 Adjoint root datum of type C2 > R1 eq R2; false %%a> assert not $1; > IsIsomorphic(R1, R2); true [ 2, 1 ] .. magma:function:: CartanName(R) :input_types: RootStr :output_types: MonStgElt :label: CartanName_RootStr The Cartan name of the root datum :math:`R` (Section :ref:`SectCartanFinAff`). .. magma:function:: TwistedCartanName(R) :input_types: RootDtm :output_types: MonStgElt :label: TwistedCartanName_RootDtm The twisted Cartan name of the root datum :math:`R`. E.g., ``"2A3,2"``. .. magma:function:: CoxeterDiagram(R) :input_types: RootStr :label: CoxeterDiagram_RootStr Print the Coxeter diagram of the root datum :math:`R` (Section :ref:`SectCartanFinAff`). .. magma:function:: DynkinDiagram(R) :input_types: RootStr :label: DynkinDiagram_RootStr Print the Dynkin diagram of the root datum :math:`R` (Section :ref:`SectCartanFinAff`). .. magma:function:: CoxeterMatrix(R) :input_types: RootStr :output_types: AlgMatElt :label: CoxeterMatrix_RootStr The Coxeter matrix of the root datum :math:`R` (Section :ref:`SectCartanCoxMat`). .. magma:function:: CoxeterGraph(R) :input_types: RootStr :output_types: GrphUnd :label: CoxeterGraph_RootStr The Coxeter graph of the root datum :math:`R` (Section :ref:`SectCartanCoxGrph`). .. magma:function:: CartanMatrix(R) :input_types: RootStr :output_types: AlgMatElt :label: CartanMatrix_RootStr The Cartan matrix of the root datum :math:`R` (Section :ref:`SectCartanCarMat`). .. magma:function:: DynkinDigraph(R) :input_types: RootStr :output_types: GrphDir :label: DynkinDigraph_RootStr The Dynkin digraph of the root datum :math:`R` (Section :ref:`SectCartanDynDigrph`). .. magma:example:: Example: Diagrams :label: Diagrams .. code-block:: magma > R := RootDatum("F4"); > DynkinDiagram(R); F4 1 - 2 =>= 3 - 4 > CoxeterDiagram(R); F4 1 - 2 === 3 - 4 .. magma:function:: GammaAction(R) :input_types: RootDtm :output_types: Rec :label: GammaAction_RootDtm The :math:`\Gamma`-action of the root datum :math:`R`. This is a record consisting of four elements: ``gamma`` is the Group :math:`\Gamma` acting on :math:`R`, ``perm_ac`` is the homomorphism defining the permutation action of :math:`\Gamma` on the set of all roots of :math:`R`, finally ``mats_rt`` and ``mats_co`` are sequences of matrices defining the action of :math:`\Gamma` on the root and coroot spaces of :math:`R`. .. magma:function:: GammaRootSpace(R) :input_types: RootDtm :output_types: GSetEnum, Map :label: GammaRootSpace_RootDtm .. magma:function:: GammaCorootSpace(R) :input_types: RootDtm :output_types: GSetEnum, Map :label: GammaCorootSpace_RootDtm Given a root datum :math:`R`, create the fixed space of :math:`\Gamma` acting on the (co)root space :math:`V = {\mathbb{Q}}\otimes X` of :math:`R` as well as the embedding in :math:`V`. .. magma:function:: GammaOrbitOnRoots(R,r) :input_types: RootDtm, RngIntElt :output_types: GSetEnum :label: GammaOrbitOnRoots_RootDtm_RngIntElt The orbit through the :math:`r`\ th root of the :math:`\Gamma`-action on the root datum :math:`R`. .. magma:function:: GammaOrbitsOnRoots(R) :input_types: RootDtm :output_types: SeqEnum[GSetEnum] :label: GammaOrbitsOnRoots_RootDtm .. magma:function:: PositiveGammaOrbitsOnRoots(R) :input_types: RootDtm :output_types: SeqEnum[GSetEnum] :label: PositiveGammaOrbitsOnRoots_RootDtm .. magma:function:: NegativeGammaOrbitsOnRoots(R) :input_types: RootDtm :output_types: SeqEnum[GSetEnum] :label: NegativeGammaOrbitsOnRoots_RootDtm .. magma:function:: ZeroGammaOrbitsOnRoots(R) :input_types: RootDtm :output_types: SeqEnum[GSetEnum] :label: ZeroGammaOrbitsOnRoots_RootDtm The sequence of all (respectively positive, negative and zero) orbits of the :math:`\Gamma`-action on the root datum :math:`R` (Section :ref:`SubsectExtRD`). .. magma:function:: GammaActionOnSimples(R) :input_types: RootDtm :output_types: HomGrp :label: GammaActionOnSimples_RootDtm The :math:`[\Gamma]`-action on the simple (co)roots of the root datum :math:`R`. (Section :ref:`SubsectExtRD`). This function was called ``GammaActionPi`` in the last release. .. magma:function:: OrbitsOnSimples(R) :input_types: RootDtm :output_types: SeqEnum[GSetEnum] :label: OrbitsOnSimples_RootDtm The sequence of all orbits of the :math:`[\Gamma]`-action on the simple (co)roots of the root datum :math:`R` (Section :ref:`SubsectExtRD`). This function was called ``OrbitsPi`` in the last release. .. magma:function:: DistinguishedOrbitsOnSimples(R) :input_types: RootDtm :output_types: SeqEnum[GSetEnum] :label: DistinguishedOrbitsOnSimples_RootDtm The sequence of distinguished orbits of the :math:`[\Gamma]`-action on the simple (co)roots of the root datum :math:`R` (Section :ref:`SubsectExtRD`). This function was called ``DistinguishedOrbitsPi`` in the last release. .. magma:function:: BaseRing(R) :input_types: RootDtm :output_types: RngInt :label: BaseRing_RootDtm The base ring of the root datum :math:`R` is the field of rational numbers. .. magma:function:: Rank(R) :input_types: RootStr :output_types: RngIntElt :label: Rank_RootStr .. magma:function:: AbsoluteRank(R) :input_types: RootDtm :output_types: RngIntElt :label: AbsoluteRank_RootDtm The (absolute) rank of the root datum :math:`R`, i.e. the number of simple (co)roots. .. magma:function:: RelativeRank(R) :input_types: RootDtm :output_types: RngIntElt :label: RelativeRank_RootDtm The relative rank of the root datum :math:`R`, i.e. the number of simple (co)roots of the relative root system. This is the same as absolute rank for split root data. .. magma:function:: Dimension(R) :input_types: RootStr :output_types: RngIntElt :label: Dimension_RootStr The dimension of the root datum :math:`R`, i.e. the dimension of the (co)root space. This is at least as large as the rank, with equality when :math:`R` is semisimple. .. magma:function:: TwistingDegree(R) :input_types: RootDtm :output_types: RngIntElt :label: TwistingDegree_RootDtm The twisting degree of the root datum :math:`R`, i.e. the order of :math:`\Gamma` divided by the kernel of the :math:`[\Gamma]`-action. .. magma:function:: AnisotropicSubdatum(R) :input_types: RootDtm :output_types: RootDtm :label: AnisotropicSubdatum_RootDtm The anisitropic subdatum of the root datum :math:`R`. .. magma:example:: Example: Operations For Twisted Root Data :label: OperationsForTwistedRootData Consider the twisted root datum of type :math:`{}^2A_{3,1}` with distinguished orbit :math:`\{2\}`: First, print out the action of :math:`\Gamma` on the root datum: Compute the orbits of the :math:`\Gamma`-action: Compute the :math:`[\Gamma]`-action and its orbits: Absolute and relative rank and the twisting degree, as well as their appearance in the name of the root datum: anisotropic subdatum: .. code-block:: magma > R := RootDatum( "A3" : Twist := < {{2}}, 2 > ); > GammaAction(R); rec | gamma := Permutation group acting on a set of cardinality 4 Order = 4 = 2^2 (1, 2, 3, 4), perm_ac := Homomorphism of GrpPerm: $, Degree 4, Order 2^2 into GrpPerm: $, Degree 12, Order 2^10 * 3^5 * 5^2 * 7 * 11 induced by (1, 2, 3, 4) |--> (1, 3, 7, 9)(2, 4, 6, 5)(8, 10, 12, 11), mats_rt := [ [ 0 0 1] [ 1 1 0] [-1 0 0] ], mats_co := [ [ 0 0 1] [ 0 1 0] [-1 1 0] ] > > PositiveGammaOrbitsOnRoots(R); [ GSet{ 2, 4, 5, 6 } ] > NegativeGammaOrbitsOnRoots(R); [ GSet{ 8, 10, 11, 12 } ] > ZeroGammaOrbitsOnRoots(R); [ GSet{ 1, 3, 7, 9 } ] > &+[ Root(R,r) : r in ZeroGammaOrbitsOnRoots(R)[1] ]; (0 0 0) > GammaActionOnSimples(R); Homomorphism of GrpPerm: $, Degree 4, Order 2^2 into GrpPerm: $, Degree 3, Order 2 * 3 induced by (1, 2, 3, 4) |--> (1, 3) > OrbitsOnSimples(R); [ GSet{ 2 }, GSet{ 1, 3 } ] > DistinguishedOrbitsOnSimples(R); [ GSet{ 2 } ] > AbsoluteRank(R); 3 %%a> assert $1 eq 3; > RelativeRank(R); 1 %%a> assert $1 eq 1; > TwistingDegree(R); 2 %%a> assert $1 eq 2; > R; R: Twisted adjoint root datum of type 2A3,1 > A := AnisotropicSubdatum(R); A; A: Twisted root datum of type 2(A1 A1)2,0 > GammaAction(A)`perm_ac; Homomorphism of GrpPerm: $, Degree 4, Order 2^2 into GrpPerm: $, Degree 4, Order 2^2 induced by (1, 2, 3, 4) |--> (1, 2, 3, 4) .. magma:function:: CoxeterGroupOrder(R) :input_types: RootStr :output_types: RngIntElt :label: CoxeterGroupOrder_RootStr The order of the (split) Coxeter group of the root datum :math:`R`. .. magma:function:: GroupOfLieTypeOrder(R, q) :input_types: RootDtm, RngElt :output_types: RngIntElt :label: GroupOfLieTypeOrder_RootDtm_RngElt The order of the group of Lie type with split root datum :math:`R` over the field of cardinality :math:`q`. .. magma:function:: GroupOfLieTypeFactoredOrder(R, q) :input_types: RootDtm, RngElt :output_types: RngIntElt :label: GroupOfLieTypeFactoredOrder_RootDtm_RngElt The factored order of the group of Lie type with split root datum :math:`R` over the field of order :math:`q`. .. magma:example:: Example: Group Of Lie Type Order :label: GroupOfLieTypeOrder As well as accepting a specific prime power, these functions also take an indeterminate so that the generic order formula can be computed. .. code-block:: magma > P := PolynomialRing(Integers()); > R := RootDatum("F4"); > GroupOfLieTypeFactoredOrder(R, q); [ , , , , , , , ] > > R := RootDatum("B2"); > ord := GroupOfLieTypeOrder(R, q); > forall{ q : q in [2..200] | not IsPrimePower(q) or > Evaluate(ord, q) eq GroupOfLieTypeOrder(R, q) }; true %%a> assert $1; .. magma:function:: FundamentalGroup(R) :input_types: RootDtm :output_types: GrpAb, Map :label: FundamentalGroup_RootDtm The fundamental group :math:`\Lambda/{\mathbb{Z}}\Phi` of the root datum :math:`R` together with the projection :math:`\Lambda\rightarrow\Lambda/{\mathbb{Z}}\Phi`. See Subsection :ref:`SubsectRDIsogeny`. .. magma:function:: IsogenyGroup(R) :input_types: RootDtm :output_types: GrpAb, Map :label: IsogenyGroup_RootDtm The isogeny group :math:`X/{\mathbb{Z}}\Phi` of the root datum :math:`R` together with the projection :math:`X\rightarrow X/{\mathbb{Z}}\Phi`. If :math:`R` is semisimple, the injection :math:`X/{\mathbb{Z}}\Phi\rightarrow\Lambda/{\mathbb{Z}}\Phi` is also returned. See Subsection :ref:`SubsectRDIsogeny`. .. magma:function:: CoisogenyGroup(R) :input_types: RootDtm :output_types: GrpAb, Map :label: CoisogenyGroup_RootDtm The coisogeny group :math:`Y/{\mathbb{Z}}\Phi^\star` of the root datum :math:`R` together with the projection :math:`Y\rightarrow Y/{\mathbb{Z}}\Phi^\star`. If :math:`R` is semisimple, the projection :math:`Y/{\mathbb{Z}}\Phi^\star\rightarrow\Lambda/{\mathbb{Z}}\Phi` is also returned. See Subsection :ref:`SubsectRDIsogeny`. .. magma:example:: Example: Isogeny Groups :label: IsogenyGroups In the semisimple case, the fundamental group contains the isogeny group, with quotient isomorphic to the coisogeny group. Nonsemisimple root data have infinite isogeny groups. .. code-block:: magma > R := RootDatum("A5" : Isogeny := 3); > F := FundamentalGroup(R); > G := IsogenyGroup(R); > H := CoisogenyGroup(R); > #G * #H eq #F; true %%a> assert $1; > R := StandardRootDatum("A", 5); > IsogenyGroup(R); Abelian Group isomorphic to Z Defined on 1 generator (free)