Homomorphisms of Group Representations

Creation of Homomorphisms between Group Representations

Homomorphism(V, W, f): ModRed, ModRed, UserProgram -> ModRedHom

Homomorphism(V, W, f): ModRed, ModRed, Map -> ModRedHom

Homomorphism(V, W, f): ModRed, ModRed, CombFreeModHom -> ModRedHom

Construct a homomorphism of group representations described by \(f : V \to W\). Does not verify that the map indeed describes a homomorphism of group representations.

Homomorphism(V, W, S): ModRed, ModRed, SeqEnum -> ModRedHom

BaseChangeCodomain: BoolElt                    Default: false

Construct a homomorphism of group representations \(f : V \to W\), mapping the basis of \(V\) to \(S\). If BaseChangeCodomain is true, returns a homomorphism \(f : V \to W \otimes R\), where \(R\) is the base ring of \(V\). Does not verify that the map indeed describes a homomorphism of group representations.

Homomorphism(M, N, f): CombFreeMod, CombFreeMod, UserProgram -> CombFreeModHom

Homomorphism(M, N, f): CombFreeMod, CombFreeMod, Map -> CombFreeModHom

Construct a homomorphism of \(R\)-modules described by \(f : M \to N\).

Homomorphism(M, N, S): CombFreeMod, CombFreeMod, SeqEnum -> CombFreeModHom

BaseChangeCodomain: BoolElt                    Default: false

Construct a homomorphism of \(R\)-modules \(f : M \to N\), mapping the basis of \(M\) to \(S\). If BaseChangeCodomain is true, returns a homomorphism \(f : M \to N \otimes R\), where \(R\) is the base ring of \(V\).

Properties of Homomorphisms of Group Representations

Domain(f): ModRedHom -> ModRed

Domain(f): CombFreeModHom -> CombFreeMod

The domain of \(f\).

Codomain(f): ModRedHom -> ModRed

Codomain(f): CombFreeModHom -> CombFreeMod

The codomain of \(f\).

Kernel(f): ModRedHom -> ModRed

The kernel of \(f\), as a representation of \(G\).

Operations on Homomorphisms of Group Representations

Evaluate(f, v): ModRedHom, ModRedElt -> ModRedElt

Evaluate(f, v): CombFreeModHom, CombFreeModElt -> CombFreeModElt

v @ f: ModRedElt, ModRedHom -> ModRedElt

v @ f: CombFreeModElt, CombFreeModHom -> CombFreeModElt

Returns \(f(v)\).

w @@ f: ModRedElt, ModRedHom -> ModRedElt

w @@ f: CombFreeModElt, CombFreeModHom -> CombFreeModElt

Given a homomorphism \(f : V \to W\), and an element \(w \in W\), returns an element \(v \in V\) such that \(f(v) = w\).