Creation of Group Representations

General Group Representations

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

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

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

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

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

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 [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

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

params: List                    Default: [* *]

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