Elements of Group Representations

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 \(v, w \in V\), return \(v + w \in V\).

v - w: ModRedElt, ModRedElt -> ModRedElt

v - w: CombFreeModElt, CombFreeModElt -> CombFreeModElt

Given \(v, w \in V\), return \(v - w \in V\).

a * v: RngElt, ModRedElt -> ModRedElt

a * v: RngElt, CombFreeModElt -> CombFreeModElt

Given \(a \in R\) and \(v \in V\), return \(av \in V\).

g * v: GrpElt, ModRedElt -> ModRedElt

Given \(g \in G\) and \(v \in V\), returns \(g(v) = g \cdot v\).

m * v: AlgMatElt, ModRedElt -> ModRedElt

Given \(v \in V \simeq R^n\) and \(m \in M_n(R) \simeq {\operatorname{End}}(V)\), return \(m(v) \in V\).

v ^ w: CombFreeModElt, CombFreeModElt -> CombFreeModElt

Given \(v, w \in \bigwedge^{\bullet} M\), return \(v \wedge w\).

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 representing \(v\).

ChangeRing(v, S): CombFreeModElt, Rng -> CombFreeModElt

Given an element \(v \in M\), where \(M\) is a combinatorial \(R\)-module, return \(v \otimes 1 \in M \otimes_R S\).