Elements of Group Representations

Contents

Creation of Elements

GroupRepresentationElement(V, m) : ModRed, CombFreeModElt -> ModRedElt
An element of the group representation V whose underlying vector is m.
CombinatorialFreeModuleElement(M, v) : CombFreeMod, ModRngElt -> CombFreeModElt
CombinatorialFreeModuleElement(M, v) : CombFreeMod, ModEDElt -> CombFreeModElt
CombinatorialFreeModuleElement(M, v) : CombFreeMod, ModTupFldElt[Fld] -> CombFreeModElt
An element of M whose underlying vector is v.

Basic Properties

Parent(v) : ModRedElt -> ModRed
Parent(v) : CombFreeModElt -> CombFreeMod
The parent of v.
ActionMatrix(V, g) : ModRed, GrpElt -> GrpMatElt
The matrix describing the action of g on V.

Operations on Elements

v + w : ModRedElt, ModRedElt -> ModRedElt
v + w : CombFreeModElt, CombFreeModElt -> CombFreeModElt
Given v, w ∈V, return v + w ∈V.
v - w : ModRedElt, ModRedElt -> ModRedElt
v - w : CombFreeModElt, CombFreeModElt -> CombFreeModElt
Given v, w ∈V, return v - w ∈V.
a * v : RngElt, ModRedElt -> ModRedElt
a * v : RngElt, CombFreeModElt -> CombFreeModElt
Given a ∈R and v ∈V, return av ∈V.
g * v : GrpElt, ModRedElt -> ModRedElt
Given g ∈G and v ∈V, returns g(v) = g .v.
m * v : AlgMatElt, ModRedElt -> ModRedElt
Given v ∈V simeq Rn and m ∈Mn(R) simeq End(V), return m(v) ∈V.
v ^ w : CombFreeModElt, CombFreeModElt -> CombFreeModElt
Given v, w ∈bigwedge M, return v ^ w.

Comparisons and Membership

v eq w : ModRedElt, ModRedElt -> BoolElt
Returns true if the elements v and w of a representation V are equal; otherwise false.
v in V : ModRedElt, ModRed -> BoolElt
v in V : CombFreeModElt, CombFreeMod -> BoolElt
Returns true if v is in the representation V; otherwise false.

Other Operations

Eltseq(v) : ModRedElt -> []
Eltseq(v) : CombFreeModElt -> []
Given an element v of a representation V, returns a sequence represeting v.
ChangeRing(v, S) : CombFreeModElt, Rng -> CombFreeModElt
Given an element v ∈M, where M is a combinatorial R-module, return v tensor 1 ∈M tensor R S.
V2.29, 21 October 2025