Morphisms of Root Data#

Morphisms are currently only defined for split root data. Let \(R_i=(X_i,\Phi_i,Y_i,\Phi_i^\star)\) be a root datum for \(i=1,2\). A morphism of root data \(\phi:R_1\to R_2\) consists of a pair of \({\mathbb{Z}}\)-linear maps \(\phi_X:X_1\to X_2\) and \(\Phi_Y:Y_1\to Y_2\) satisfying

  1. \(\phi_X(\Phi_1)\subseteq\Phi_2\cup\{0\}\); and

  2. \(\phi_Y(\alpha^\star)=\phi_X(\alpha)^\star\) (with the convention that \(0^\star=0\)).

A fractional morphism is similar, except that it consists of \({\mathbb{Q}}\)-linear maps on the (co)root spaces \(X_1\otimes{\mathbb{Q}}\to X_2\otimes{\mathbb{Q}}\) and \(Y_1\otimes{\mathbb{Q}}\to Y_2\otimes{\mathbb{Q}}\). The main examples of fractional morphisms are isogeny maps (Section Isogeny of Split Reduced Root Data). A dual morphism is similar, except that the maps are \(X_1\to Y_2\) and \(Y_1\to X_2\). This is clearly equivalent to a morphism from \(R_1\) to the dual of \(R_2\). Finally we define a dual fractional morphism in the obvious way.

A (fractional) morphism \(\phi:R_1\to R_2\) also stores a sign corresponding to each simple root of \(R_1\). This has no effect on the action of \(\phi\) on roots or coroots, but does effect the definition of the corresponding homomorphisms of Lie algebras and groups of Lie type.

hom<R -\>S | phiX, phiY>: RootDtm, RootDtm, Map, Map -> Map#
hom<R -\>S | phiX, phiY>: RootDtm, RootDtm, Mtrx, Mtrx -> Map#

Construct a (fractional) morphism \(\phi : R\to S\) of root data with the given linear maps or matrices phiX and phiY for the action of \(\phi\) on \(X_1\) and \(Y_1\).

hom<R -\>S | Q>: RootDtm, RootDtm, [RngIntElt] -> Map#

Construct a (fractional) morphism of root data \(R\to S\) with the given sequence of root images. The sequence \(Q\) must have length \(2N\) and consist of elements in the range \([0,\dots,2M]\), where \(N\) is the number of positive roots of \(R\) and \(M\) is the number of positive roots of \(S\). The domain \(R\) must be semisimple.

Morphism(R, S, phiX, phiY): RootDtm, RootDtm, Map, Map -> Map#
Morphism(R, S, phiX, phiY): RootDtm, RootDtm, Mtrx, Mtrx -> Map#
Check: BoolElt                    Default: true

Construct a (fractional) morphism \(\phi : R\to S\) of root data with the given linear maps or matrices phiX and phiY for the action of \(\phi\) on \(X_1\) and \(Y_1\). The domain \(R\) must be semisimple.

If Check is set to false, the function does not check that the maps send (co)roots to (co)roots. This function is the same as the constructor hom, except for the optional parameter.

Morphism(R, S, Q): RootDtm, RootDtm, [RngIntElt] -> Map#
Check: BoolElt                    Default: true

Construct a (fractional) morphism of root data \(R\to S\) with the given sequence of root images. The sequence \(Q\) must have length \(2N\) and consist of elements in the range \([0,\dots,2M]\), where \(N\) is the number of positive roots of \(R\) and \(M\) is the number of positive roots of \(S\). The domain \(R\) must be semisimple.

If Check is set to false, the function does not check that the maps send (co)roots to (co)roots. This function is the same as the constructor hom, except for the optional parameter.

DualMorphism(R, S, phiX, phiY): RootDtm, RootDtm, Map, Map -> Map#
DualMorphism(R, S, phiX, phiY): RootDtm, RootDtm, Mtrx, Mtrx -> Map#
Check: BoolElt                    Default: true

Construct a (fractional) dual morphism of root data \(R\to S\) with the given linear maps or matrices of linear maps. If Check is set to false, the function does not check that the maps send (co)roots to (co)roots.

DualMorphism(R, S, Q): RootDtm, RootDtm, [RngIntElt] -> Map#
Check: BoolElt                    Default: true

Construct a (fractional) dual morphism of root data \(R\to S\) with the given sequence of root images. The sequence \(Q\) must have length \(2N\) and consist of elements in the range \([0,\dots,2M]\), where \(N\) is the number of positive roots of \(R\) and \(M\) is the number of positive roots of \(S\). The domain \(R\) must be semisimple. If Check is set to false, the function does not check that the maps send (co)roots to (co)roots.

RootImages(phi): Map -> [RngIntElt]#

The indices of the root images of the (dual) (fractional) morphism \(\phi\).

RootPermutation(phi): Map -> GrpPermElt#

The indices of the root images of the automorphism \(\phi\).

IdentityMap(R): RootDtm -> Map#
IdentityAutomorphism(R): RootDtm -> Map#

The identity morphism \(R\to R\).

Example: Creating Root Data Homomorphisms (ex-b874c1)#

We construct the fractional morphism from the standard root datum of type \(A_3\) onto the adjoint root datum of type \(A_3\). This will allow us to construct the algebraic projection \({GL}_4\to{PGL}_4\) in Section Algebraic Homomorphisms.

> RGL := StandardRootDatum( "A", 3 );
> RPGL := RootDatum( "A3" );
> A := VerticalJoin( SimpleRoots(RGL), Vector([Rationals()|1,1,1,1]) )^-1 *
>   VerticalJoin( SimpleRoots(RPGL), Vector([Rationals()|0,0,0]) );
> B := VerticalJoin( SimpleCoroots(RGL), Vector([Rationals()|1,1,1,1]) )^-1 *
>   VerticalJoin( SimpleCoroots(RPGL), Vector([Rationals()|0,0,0]) );
> phi := hom< RGL -> RPGL | A, B >;
> v := Coroot(RGL,1);
> v; phi(v);
( 1 -1  0  0)
( 2 -1  0 )

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