# Creation of Group Representations

## General Group Representations

### `GroupRepresentation(G, M, action): Grp, CombFreeMod, MonStgElt -> ModRed`

### `GroupRepresentation(G, M, action): GrpRed, CombFreeMod, MonStgElt -> ModRed`

```magma
params: List                    Default: [* *]
```

Constructs a group representation for the group $G$ on the combinatorial free module $M$ with basis $B$, such that the action on basis elements $G \times B \to M$ is described by the map given by the string `action`.

## Subrepresentations

### `Subrepresentation(V, t): ModRed, Any -> ModRed, ModRedHom`

### `SubCFModule(V, t): CombFreeMod, Any -> CombFreeMod, CombFreeModHom`

The subrepresentation (or submodule) of $V$ whose underlying free module is generated by $t$.

## Natural Representations

### `TrivialRepresentation(G, R): Grp, Rng -> ModRed`

### `TrivialRepresentation(G, R): GrpRed, Rng -> ModRed`

```magma
name: MonStgElt                    Default: "v"
```

The trivial representation for the group $G$ over the ring $R$, where the basis element has name `name`.

### `StandardRepresentation(G): GrpMat -> ModRed`

### `StandardRepresentation(G): GrpRed -> ModRed`

```magma
name: MonStgElt                    Default: "x"
```

The standard representation of the matrix group $G$ over its ring of definition $R$, i.e. the representation obtained by considering its given embedding in ${\operatorname{GL}}_n(R)$ acting on $R^n$ by invertible linear transformations. The basis will have names $x_1, \ldots, x_n$, where $x$ is specified by `name`.

### `SpinorNormRepresentation(G, d): GrpRed, RngIntElt -> ModRed`

```magma
name: MonStgElt                    Default: "x"
```

The $1$-dimensional representation ${\rm spin}_d$ of the orthogonal group $G$ induced by the spinor norm and $d$.

### `Rho(G, k, j): GrpMat, RngIntElt, RngIntElt -> ModRed`

The representation $\det^k \otimes {\operatorname{Sym}}^j$.

### `SymSpinor(G, d, k): GrpRed, RngIntElt, RngIntElt -> ModRed`

The representation ${\rm spin}_d \otimes {\operatorname{Sym}}^k$ of the orthogonal group $G$.

### `AltSpinor(G, d): GrpRed, RngIntElt, RngIntElt -> ModRed`

The representation ${\rm spin}_d \otimes {\operatorname{Alt}}^j$ of the orthogonal group $G$.

### `RadicalSignCharacterSinglePrime(G, p): GrpRed, RngIntElt -> ModRed`

The character $\theta_p$ from [[Dummigan *et al.*, 2024](../../references.md#cite-dprt24)]. If $G = {\rm O}(Q)$ is an orthogonal group defined over ${\mathbb{Q}}$, with $Q$ integral, and $p$ is a prime divisor of ${\rm Disc}(Q)$, returns the $1$-dimensional representation of $G({\mathbb{Z}}_p)$ on the determinant of the radical of $Q$ mod $2p$.

### `RadicalSignCharacter(G, d): GrpRed, RngIntElt -> ModRed`

The character $\theta_d = \prod_{p \mid d} \theta_p$, where $\theta_p$ is the character constructed by the function `RadicalSignCharacterSinglePrime`.

### `SpinRepresentation(G, p): GrpRed, RngIntElt -> ModRed`

The spin representation of the special orthogonal group $G={\operatorname{SO}}(Q)$ with coefficients in ${\mathbb{F}}_p$.

## New Representations from Old

### `DeterminantRepresentation(G): GrpMat -> ModRed`

```magma
k   : RngIntElt                    Default: 1
name: MonStgElt                    Default: "v"
```

The $1$-dimensional representation, where $G$ acts via the $g \mapsto \det(g)^k$, where the basis element has name `name`.

### `SymmetricRepresentation(V, n): ModRed, RngIntElt -> ModRed`

The symmetric representation ${\operatorname{Sym}}^n(V)$.

### `AlternatingRepresentation(V, n): ModRed, RngIntElt -> ModRed`

The alternating representation ${\operatorname{Alt}}^n(V)$.

### `DualRepresentation(V): ModRed -> ModRed`

The dual (contragredient) representation $V^{\vee}$.

### `TensorProduct(V, W): ModRed, ModRed -> ModRed`

The representation $V \otimes W$ with the diagonal action.

### `TensorPower(V, d): ModRed, RngIntElt -> ModRed`

The tensor power representation $V^{\otimes d}$, with the diagonal action.

### `Pullback(V, f, G): ModRed, MonStgElt, Grp -> ModRed`

If $f : G \to H$ is a group homomorphism, and $V$ is a representation of $H$, returns the pullback of $V$ via $f$ to a representation of $G$. Does not verify that $f$ is a group homomorphism.

## New Combinatorially Free Modules from Old

### `ExteriorPower(M, n): CombFreeMod, RngIntElt -> CombFreeMod`

### `AlternatingPower(M, n): CombFreeMod, RngIntElt -> CombFreeMod`

Returns $\bigwedge^n M$.

### `ExteriorAlgebra(M): CombFreeMod -> CombFreeMod, CombFreeModHom`

The underlying module of the exterior algebra of $M$, namely $\bigwedge^{\bullet} M$, together with an embedding of $M$ as the degree $1$ component.

### `DirectSum(M): [ CombFreeMod ] -> CombFreeMod`

The direct sum $\bigoplus_i M_i$.

## Highest Weight Representations

### `GroupRepresentation(G, w): GrpLie, [ RngIntElt ] -> ModRed`

The algebraic representation of the group $G$ with highest weight $w$. Embeds the group of Lie type $G$ into its standard representation.

### `HighestWeightRepresentation(G, w): GrpRed, [ RngIntElt ] -> ModRed`

The irreducible algebraic representation of $G$ with highest weight $w$.

### `HighestWeightRepresentation(G, w, p): GrpRed, [ RngIntElt ], RngIntElt -> ModRed`

If $G$ is a reductive group defined over a number field $F$, return the irreducible algebraic representation of $G$ with highest weight $w$, and coefficient field of characteristic $p$, obtained by the reduction modulo a prime $P$ of $F$ above $p$.

## Creation of Combinatorial Free Modules

### `CombinatorialFreeModule(R, S): Rng, SetIndx -> CombFreeMod`

### `CombinatorialFreeModule(R, S): Rng, [ MonStgElt ] -> CombFreeMod`

```magma
params: List                    Default: [* *]
```

A combinatorial free module over the ring $R$ with basis given by $S$, namely $M = R^S$.