.. _SectRDSub: .. _create-new-root-datum: Building Root Data ================== .. magma:constructor:: sub :input_types: RootDtm, SetEnum :output_types: RootDtm The root subdatum of the root datum :math:`R` 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. .. magma:constructor:: sub :input_types: RootDtm, SetEnum :output_types: RootDtm The root subdatum of the root datum :math:`R` generated by the 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 (i.e. none of them may be a summand of another) otherwise an error is signalled. The simple roots will appear in the subdatum in the given order. .. magma:example:: Example: Root Subdata :label: RootSubdata .. code-block:: magma > R := RootDatum("A4"); > PositiveRoots(R); {@ (1 0 0 0), (0 1 0 0), (0 0 1 0), (0 0 0 1), (1 1 0 0), (0 1 1 0), (0 0 1 1), (1 1 1 0), (0 1 1 1), (1 1 1 1) @} > s := sub< R | [6,1,4] >; > s; Root datum of type A3 > PositiveRoots(s); {@ (0 1 1 0), (1 0 0 0), (0 0 0 1), (1 1 1 0), (0 1 1 1), (1 1 1 1) @} \; Error: The given roots are not simple in a subdatum > s := sub< R | {1,5} >; > s; Root datum of type A2 > PositiveRoots(s); {@ (1 0 0 0), (0 1 0 0), (1 1 0 0) @} .. magma:operation:: R1 subset R2 :input_types: RootDtm, RootDtm :output_types: BoolElt, . :label: op_subset_RootDtm_RootDtm Returns ``true`` if and only if the root datum :math:`R_1` is a subset of the root datum :math:`R_2`. If true, returns an injection as sequence of roots as second return value. .. magma:operation:: R1 + R2 :input_types: RootDtm, RootDtm :output_types: RootDtm :label: op_plus_RootDtm_RootDtm .. magma:function:: DirectSum(R1, R2) :input_types: RootDtm, RootDtm :output_types: RootDtm :label: DirectSum_RootDtm_RootDtm The external direct sum of the root data :math:`R_1` and :math:`R_2`. The full (co)root space of the result is the direct sum of the full (co)root spaces of :math:`R_1` and :math:`R_2`. .. magma:operation:: R1 join R2 :input_types: RootDtm, RootDtm :output_types: RootDtm :label: op_join_RootDtm_RootDtm The internal direct sum of the root data :math:`R_1` and :math:`R_2`. The root data must have the same full (co)root space, which will also be the full (co)root space of the result. The root data must have disjoint (co)root spaces. .. magma:example:: Example: Root Dtm Sums :label: RootDtmSums .. code-block:: magma > R := RootDatum("A1A1"); > R1 := sub; > R2 := sub; > R1 + R2; Root datum of dimension 4 of type A1 A1 > R1 join R2; R: Adjoint root datum of dimension 2 of type A1 A1 .. magma:function:: DirectSumDecomposition(R) :input_types: RootDtm :output_types: [], RootDtm, Map :label: DirectSumDecomposition_RootDtm .. magma:function:: IndecomposableSummands(R) :input_types: RootDtm :output_types: [], RootDtm, Map :label: IndecomposableSummands_RootDtm Returns a sequence :math:`Q` of irreducible root data, a root datum :math:`S` which is the direct sum of the terms of :math:`Q`, and an isogeny map :math:`\phi:S\to R`. The root datum :math:`R` must be semisimple. Note that a semisimple root datum :math:`R` need not be a direct sum of simple root data, but it is isogenous to a direct sum of root data :math:`S`. .. magma:example:: Example: Root Dtm Decomp :label: RootDtmDecomp If the root datum in adjoint or simply connected, then it is a direct sum of simples. In this case we get :math:`S=R`. The join of the summands of the direct sum decomposition is the original root datum again: Here is an example of a semisimple root datum which is not a direct sum of simple subdata. Note that a simple root datum of type :math:`A_1` is either simply connected or adjoint. .. code-block:: magma > R := RootDatum("A4B5" : Isogeny:="SC"); > Q, S := DirectSumDecomposition( R ); > R eq S; true %%a> assert $1; > R eq Q[1] join Q[2]; true %%a> assert $1; > R eq &join DirectSumDecomposition(R); true %%a> assert $1; > R eq &+ DirectSumDecomposition(R); false %%a> assert not $1; > R1 := RootDatum("A3T2B4T3"); > R2 := RootDatum("T3G2T4BC3"); > R1 + R2; Adjoint root datum of dimension 24 of type A3 B4 G2 BC3 > R1 join R2; Root datum of dimension 12 of type A3 B4 G2 BC3 > G:=FundamentalGroup("A1A1"); > _,inj:=sub; > R:=RootDatum("A1A1":Isogeny:=inj); > ad := RootDatum( "A1" : Isogeny:="Ad" ); > sc := RootDatum( "A1" : Isogeny:="SC" ); > IsIsomorphic( R, DirectSum(ad,ad) ); false %%a> assert not $1; > IsIsomorphic( R, DirectSum(ad,sc) ); false %%a> assert not $1; > IsIsomorphic( R, DirectSum(sc,sc) ); false %%a> assert not $1; > Q, S := DirectSumDecomposition( R ); > R eq S; false %%a> assert not $1; .. magma:function:: Dual(R) :input_types: RootDtm :output_types: RootDtm, Map :label: Dual_RootDtm The dual of the root datum :math:`R`, obtained by swapping the roots and coroots. The second value returned is the dual morphism from :math:`R` to its dual. .. magma:function:: SimplyConnectedVersion(R) :input_types: RootDtm :output_types: RootDtm, Map :label: SimplyConnectedVersion_RootDtm The simply connected version of the root datum :math:`R`. If :math:`R` is semisimple then the injection of the simply connected version into :math:`R` is returned as the second value. .. magma:function:: AdjointVersion(R) :input_types: RootDtm :output_types: RootDtm, Map :label: AdjointVersion_RootDtm The adjoint version of the root datum :math:`R`. If :math:`R` is semisimple then the projection from :math:`R` to its adjoint version is returned as the second value. .. magma:function:: IndivisibleSubdatum(R) :input_types: RootDtm :output_types: RootDtm :label: IndivisibleSubdatum_RootDtm The root datum consisting of all indivisible roots of the root datum :math:`R`. .. magma:function:: Radical(R) :input_types: RootDtm :output_types: RootDtm :label: Radical_RootDtm The radical of the root datum :math:`R`, ie, the toral subdatum whose root (resp. coroot) space consists of the vectors perpendicular to every coroot (resp. root). .. magma:example:: Example: Direct Sum Dual Radical :label: DirectSumDualRadical An adjoint or simply connected root datum is always a direct sum of irreducible subdata. In these cases we take :math:`S=R`. .. code-block:: magma > R1 := RootDatum("A5"); > R2 := RootDatum("B4"); > R := DirectSum(R1, Dual(R2)); > DirectSumDecomposition(R); { Root datum of type A5 , Root datum of type C4 } > R := RootDatum("BC2"); > I := IndivisibleSubdatum(R); I; I: Root datum of type B2 > I subset R; true [ 1, 2, 3, 5, 7, 8, 9, 11 ] > R := StandardRootDatum("A", 3); > Radical(R); Toral root datum of dimension 1 .. magma:function:: TwistedRootDatum(R) :input_types: RootDtm :output_types: RootDtm :label: TwistedRootDatum_RootDtm .. magma:function:: TwistedRootDatum(N) :input_types: MonStgElt :output_types: RootDtm :label: TwistedRootDatum_MonStgElt :parameters: Twist : Any : 1 Create a twisted root datum from the root datum :math:`R`, or from the semisimple root datum with Cartan name :math:`N`. The twist may be specified in any of the following ways: :math:`\bullet` An integer, specifying the order of the twist; :math:`\bullet` A permutation, specifying the action of the primitive roots; :math:`\bullet` A pair :math:``, where :math:`D` is a set of distinguished orbits as sets of integers, and :math:`i` is the order of the Dynkin diagram symmetry; :math:`\bullet` A pair :math:`<\Gamma, Q>`, where :math:`\Gamma` is the acting group, and :math:`Q` is a sequence containing the permutation of the primitive roots for each of the generators of :math:`\Gamma`; :math:`\bullet` A homomorphism from :math:`\Gamma` to the symmetric group whose order is the number of roots of :math:`R`, describing how the acting group :math:`\Gamma` acts on the roots. .. magma:example:: Example: Direct Sum Dual Radical :label: DirectSumDualRadical We construct a twisted root datum in a number of ways. .. code-block:: magma > S := TwistedRootDatum("D4" : Twist := 3); > S; S: Twisted adjoint root datum of dimension 4 of type 3D4,2 \bln > R := RootDatum("A1A3"); > DynkinDiagram(R); \bln A1 1 \bln A3 2 - 3 - 4 > S := TwistedRootDatum(R : Twist := Sym(4)!(2,4)); > S; S: Twisted adjoint root datum of dimension 4 of type 2(A1 A3)4,3 \bln > S := TwistedRootDatum("A4" : Twist := <{{1,4},{2,3}}, 2>); > S; S: Twisted adjoint root datum of dimension 4 of type 2A4,2 \bln > R := RootDatum("E6" : Isogeny := "SC"); > DynkinDiagram(R); \bln E6 1 - 3 - 4 - 5 - 6 | 2 > S := TwistedRootDatum(R : Twist := ); > S; S: Twisted simply connected root datum of dimension 6 of type 2E6,4 \bln > R := RootDatum("D4"); > DynkinDiagram(R); \bln D4 3 / 1 - 2 \\ 4 > Gamma := Sym(3); > Gamma.1, Gamma.2; (1, 2, 3) (1, 2) > S := TwistedRootDatum(R : Twist := ); > S; S: Twisted adjoint root datum of dimension 4 of type 6D4,2 \bln > R := RootDatum("A2"); > DynkinDiagram(R); \bln A2 1 - 2 > Roots(R); {@ (1 0), (0 1), (1 1), (-1 0), ( 0 -1), (-1 -1) @} > S6 := Sym(#Roots(R)); > phi := hom S6 | S6!(1,2)(4,5)>; > S := TwistedRootDatum(R : Twist := phi); .. magma:function:: UntwistedRootDatum(R) :input_types: RootDtm :output_types: RootDtm :label: UntwistedRootDatum_RootDtm .. magma:function:: SplitRootDatum(R) :input_types: RootDtm :output_types: RootDtm :label: SplitRootDatum_RootDtm The split version of the (twisted) root datum :math:`R`.