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 \(v \otimes 1 \in M \otimes_R S\).