# 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 $v \otimes 1 \in M \otimes_R S$.