Homomorphisms of Group Representations

Contents

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 tensor 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 tensor 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 ∈W, returns an element v ∈V such that f(v) = w.
V2.29, 21 October 2025