# Group Representations

- [Introduction](introduction.md)

  - [Definitions and Background](introduction.md#definitions-and-background)

  - [Categories](introduction.md#categories)

- [Creation of Group Representations](creation.md)

  - [General Group Representations](creation.md#general-group-representations)

    - [`GroupRepresentation(G, M, action): Grp, CombFreeMod, MonStgElt → ModRed`](creation.md#function-grouprepresentation-grp-combfreemod-monstgelt)

    - [`GroupRepresentation(G, M, action): GrpRed, CombFreeMod, MonStgElt → ModRed`](creation.md#function-grouprepresentation-grpred-combfreemod-monstgelt)

  - [Subrepresentations](creation.md#subrepresentations)

    - [`Subrepresentation(V, t): ModRed, Any → ModRed, ModRedHom`](creation.md#function-subrepresentation-modred-any)

    - [`SubCFModule(V, t): CombFreeMod, Any → CombFreeMod, CombFreeModHom`](creation.md#function-subcfmodule-combfreemod-any)

  - [Natural Representations](creation.md#natural-representations)

    - [`TrivialRepresentation(G, R): Grp, Rng → ModRed`](creation.md#function-trivialrepresentation-grp-rng)

    - [`TrivialRepresentation(G, R): GrpRed, Rng → ModRed`](creation.md#function-trivialrepresentation-grpred-rng)

    - [`StandardRepresentation(G): GrpMat → ModRed`](creation.md#function-standardrepresentation-grpmat)

    - [`StandardRepresentation(G): GrpRed → ModRed`](creation.md#function-standardrepresentation-grpred)

    - [`SpinorNormRepresentation(G, d): GrpRed, RngIntElt → ModRed`](creation.md#function-spinornormrepresentation-grpred-rngintelt)

    - [`Rho(G, k, j): GrpMat, RngIntElt, RngIntElt → ModRed`](creation.md#function-rho-grpmat-rngintelt-rngintelt)

    - [`SymSpinor(G, d, k): GrpRed, RngIntElt, RngIntElt → ModRed`](creation.md#function-symspinor-grpred-rngintelt-rngintelt)

    - [`AltSpinor(G, d): GrpRed, RngIntElt, RngIntElt → ModRed`](creation.md#function-altspinor-grpred-rngintelt-rngintelt)

    - [`RadicalSignCharacterSinglePrime(G, p): GrpRed, RngIntElt → ModRed`](creation.md#function-radicalsigncharactersingleprime-grpred-rngintelt)

    - [`RadicalSignCharacter(G, d): GrpRed, RngIntElt → ModRed`](creation.md#function-radicalsigncharacter-grpred-rngintelt)

    - [`SpinRepresentation(G, p): GrpRed, RngIntElt → ModRed`](creation.md#function-spinrepresentation-grpred-rngintelt)

  - [New Representations from Old](creation.md#new-representations-from-old)

    - [`DeterminantRepresentation(G): GrpMat → ModRed`](creation.md#function-determinantrepresentation-grpmat)

    - [`SymmetricRepresentation(V, n): ModRed, RngIntElt → ModRed`](creation.md#function-symmetricrepresentation-modred-rngintelt)

    - [`AlternatingRepresentation(V, n): ModRed, RngIntElt → ModRed`](creation.md#function-alternatingrepresentation-modred-rngintelt)

    - [`DualRepresentation(V): ModRed → ModRed`](creation.md#function-dualrepresentation-modred)

    - [`TensorProduct(V, W): ModRed, ModRed → ModRed`](creation.md#function-tensorproduct-modred-modred)

    - [`TensorPower(V, d): ModRed, RngIntElt → ModRed`](creation.md#function-tensorpower-modred-rngintelt)

    - [`Pullback(V, f, G): ModRed, MonStgElt, Grp → ModRed`](creation.md#function-pullback-modred-monstgelt-grp)

  - [New Combinatorially Free Modules from Old](creation.md#new-combinatorially-free-modules-from-old)

    - [`ExteriorPower(M, n): CombFreeMod, RngIntElt → CombFreeMod`](creation.md#function-exteriorpower-combfreemod-rngintelt)

    - [`AlternatingPower(M, n): CombFreeMod, RngIntElt → CombFreeMod`](creation.md#function-alternatingpower-combfreemod-rngintelt)

    - [`ExteriorAlgebra(M): CombFreeMod → CombFreeMod, CombFreeModHom`](creation.md#function-exterioralgebra-combfreemod)

    - [`DirectSum(M): [ CombFreeMod ] → CombFreeMod`](creation.md#function-directsum-combfreemod)

  - [Highest Weight Representations](creation.md#highest-weight-representations)

    - [`GroupRepresentation(G, w): GrpLie, [ RngIntElt ] → ModRed`](creation.md#function-grouprepresentation-grplie-rngintelt)

    - [`HighestWeightRepresentation(G, w): GrpRed, [ RngIntElt ] → ModRed`](creation.md#function-highestweightrepresentation-grpred-rngintelt)

    - [`HighestWeightRepresentation(G, w, p): GrpRed, [ RngIntElt ], RngIntElt → ModRed`](creation.md#function-highestweightrepresentation-grpred-rngintelt-rngintelt)

  - [Creation of Combinatorial Free Modules](creation.md#creation-of-combinatorial-free-modules)

    - [`CombinatorialFreeModule(R, S): Rng, SetIndx → CombFreeMod`](creation.md#function-combinatorialfreemodule-rng-setindx)

    - [`CombinatorialFreeModule(R, S): Rng, [ MonStgElt ] → CombFreeMod`](creation.md#function-combinatorialfreemodule-rng-monstgelt)

- [Basic Properties](basic-properties.md)

  - [Accessing Representation Information](basic-properties.md#accessing-representation-information)

    - [`Rank(V): ModRed → RngIntElt`](basic-properties.md#function-rank-modred)

    - [`Dimension(V): ModRed → RngIntElt`](basic-properties.md#function-dimension-modred)

    - [`Ngens(V): ModRed → RngIntElt`](basic-properties.md#function-ngens-modred)

    - [`Basis(V): ModRed → [ ModRedElt ]`](basic-properties.md#function-basis-modred)

    - [`BaseRing(V): ModRed → Rng`](basic-properties.md#function-basering-modred)

    - [`Group(V): ModRed → Grp`](basic-properties.md#function-group-modred)

    - [`CFM(V): ModRed → CombFreeMod`](basic-properties.md#function-cfm-modred)

  - [Accessing Combinatorial Free Module Information](basic-properties.md#accessing-combinatorial-free-module-information)

    - [`Rank(M): CombFreeMod → RngIntElt`](basic-properties.md#function-rank-combfreemod)

    - [`Dimension(M): CombFreeMod → RngIntElt`](basic-properties.md#function-dimension-combfreemod)

    - [`Ngens(M): CombFreeMod → RngIntElt`](basic-properties.md#function-ngens-combfreemod)

    - [`Basis(M): CombFreeMod → [ CombFreeModElt ]`](basic-properties.md#function-basis-combfreemod)

    - [`BaseRing(M): CombFreeMod → Rng`](basic-properties.md#function-basering-combfreemod)

    - [`Names(M): CombFreeMod → SetIndx`](basic-properties.md#function-names-combfreemod)

    - [`Names(M): CombFreeMod → [ MonStgElt ]`](basic-properties.md#function-names-combfreemod-2)

  - [Predicates](basic-properties.md#predicates)

    - [`IsTrivial(V): ModRed → BoolElt`](basic-properties.md#function-istrivial-modred)

- [Operations on Group Representations](ops.md)

  - [`Intersection(V, W): ModRed, ModRed → ModRed`](ops.md#function-intersection-modred-modred)

  - [`V meet W: ModRed, ModRed → ModRed`](ops.md#operation-op-meet-modred-modred)

  - [`V eq W: ModRed, ModRed → BoolElt`](ops.md#operation-op-eq-modred-modred)

  - [`V eq W: CombFreeMod, CombFreeMod → BoolElt`](ops.md#operation-op-eq-combfreemod-combfreemod)

  - [Base Change](ops.md#base-change)

    - [`ChangeRing(V, S): ModRed, Rng → ModRed`](ops.md#function-changering-modred-rng)

    - [`ChangeRing(M, S): CombFreeMod, Rng → CombFreeMod`](ops.md#function-changering-combfreemod-rng)

  - [Other Operations](ops.md#other-operations)

    - [`FixedSubspace(H, V): GrpMat, ModRed → ModRed`](ops.md#function-fixedsubspace-grpmat-modred)

- [Elements of Group Representations](elements.md)

  - [Creation of Elements](elements.md#creation-of-elements)

    - [`GroupRepresentationElement(V, m): ModRed, CombFreeModElt → ModRedElt`](elements.md#function-grouprepresentationelement-modred-combfreemodelt)

    - [`CombinatorialFreeModuleElement(M, v): CombFreeMod, ModRngElt → CombFreeModElt`](elements.md#function-combinatorialfreemoduleelement-combfreemod-modrngelt)

    - [`CombinatorialFreeModuleElement(M, v): CombFreeMod, ModEDElt → CombFreeModElt`](elements.md#function-combinatorialfreemoduleelement-combfreemod-modedelt)

    - [`CombinatorialFreeModuleElement(M, v): CombFreeMod, ModTupFldElt[Fld] → CombFreeModElt`](elements.md#function-combinatorialfreemoduleelement-combfreemod-modtupfldelt-fld)

  - [Basic Properties](elements.md#basic-properties)

    - [`Parent(v): ModRedElt → ModRed`](elements.md#function-parent-modredelt)

    - [`Parent(v): CombFreeModElt → CombFreeMod`](elements.md#function-parent-combfreemodelt)

    - [`ActionMatrix(V, g): ModRed, GrpElt → GrpMatElt`](elements.md#function-actionmatrix-modred-grpelt)

  - [Operations on Elements](elements.md#operations-on-elements)

    - [`v + w: ModRedElt, ModRedElt → ModRedElt`](elements.md#operation-op-plus-modredelt-modredelt)

    - [`v + w: CombFreeModElt, CombFreeModElt → CombFreeModElt`](elements.md#operation-op-plus-combfreemodelt-combfreemodelt)

    - [`v - w: ModRedElt, ModRedElt → ModRedElt`](elements.md#operation-op-minus-modredelt-modredelt)

    - [`v - w: CombFreeModElt, CombFreeModElt → CombFreeModElt`](elements.md#operation-op-minus-combfreemodelt-combfreemodelt)

    - [`a * v: RngElt, ModRedElt → ModRedElt`](elements.md#operation-op-times-rngelt-modredelt)

    - [`a * v: RngElt, CombFreeModElt → CombFreeModElt`](elements.md#operation-op-times-rngelt-combfreemodelt)

    - [`g * v: GrpElt, ModRedElt → ModRedElt`](elements.md#operation-op-times-grpelt-modredelt)

    - [`m * v: AlgMatElt, ModRedElt → ModRedElt`](elements.md#operation-op-times-algmatelt-modredelt)

    - [`v ^ w: CombFreeModElt, CombFreeModElt → CombFreeModElt`](elements.md#operation-op-pow-combfreemodelt-combfreemodelt)

  - [Comparisons and Membership](elements.md#comparisons-and-membership)

    - [`v eq w: ModRedElt, ModRedElt → BoolElt`](elements.md#operation-op-eq-modredelt-modredelt)

    - [`v in V: ModRedElt, ModRed → BoolElt`](elements.md#operation-op-in-modredelt-modred)

    - [`v in V: CombFreeModElt, CombFreeMod → BoolElt`](elements.md#operation-op-in-combfreemodelt-combfreemod)

  - [Other Operations](elements.md#other-operations)

    - [`Eltseq(v): ModRedElt → []`](elements.md#function-eltseq-modredelt)

    - [`Eltseq(v): CombFreeModElt → []`](elements.md#function-eltseq-combfreemodelt)

    - [`ChangeRing(v, S): CombFreeModElt, Rng → CombFreeModElt`](elements.md#function-changering-combfreemodelt-rng)

- [Homomorphisms of Group Representations](morphisms.md)

  - [Creation of Homomorphisms between Group Representations](morphisms.md#creation-of-homomorphisms-between-group-representations)

    - [`Homomorphism(V, W, f): ModRed, ModRed, UserProgram → ModRedHom`](morphisms.md#function-homomorphism-modred-modred-userprogram)

    - [`Homomorphism(V, W, f): ModRed, ModRed, Map → ModRedHom`](morphisms.md#function-homomorphism-modred-modred-map)

    - [`Homomorphism(V, W, f): ModRed, ModRed, CombFreeModHom → ModRedHom`](morphisms.md#function-homomorphism-modred-modred-combfreemodhom)

    - [`Homomorphism(V, W, S): ModRed, ModRed, SeqEnum → ModRedHom`](morphisms.md#function-homomorphism-modred-modred-seqenum)

    - [`Homomorphism(M, N, f): CombFreeMod, CombFreeMod, UserProgram → CombFreeModHom`](morphisms.md#function-homomorphism-combfreemod-combfreemod-userprogram)

    - [`Homomorphism(M, N, f): CombFreeMod, CombFreeMod, Map → CombFreeModHom`](morphisms.md#function-homomorphism-combfreemod-combfreemod-map)

    - [`Homomorphism(M, N, S): CombFreeMod, CombFreeMod, SeqEnum → CombFreeModHom`](morphisms.md#function-homomorphism-combfreemod-combfreemod-seqenum)

  - [Properties of Homomorphisms of Group Representations](morphisms.md#properties-of-homomorphisms-of-group-representations)

    - [`Domain(f): ModRedHom → ModRed`](morphisms.md#function-domain-modredhom)

    - [`Domain(f): CombFreeModHom → CombFreeMod`](morphisms.md#function-domain-combfreemodhom)

    - [`Codomain(f): ModRedHom → ModRed`](morphisms.md#function-codomain-modredhom)

    - [`Codomain(f): CombFreeModHom → CombFreeMod`](morphisms.md#function-codomain-combfreemodhom)

    - [`Kernel(f): ModRedHom → ModRed`](morphisms.md#function-kernel-modredhom)

  - [Operations on Homomorphisms of Group Representations](morphisms.md#operations-on-homomorphisms-of-group-representations)

    - [`Evaluate(f, v): ModRedHom, ModRedElt → ModRedElt`](morphisms.md#function-evaluate-modredhom-modredelt)

    - [`Evaluate(f, v): CombFreeModHom, CombFreeModElt → CombFreeModElt`](morphisms.md#function-evaluate-combfreemodhom-combfreemodelt)

    - [`v @ f: ModRedElt, ModRedHom → ModRedElt`](morphisms.md#operation-operation-modredelt-modredhom-modredelt)

    - [`v @ f: CombFreeModElt, CombFreeModHom → CombFreeModElt`](morphisms.md#operation-operation-combfreemodelt-combfreemodhom-combfreemodelt)

    - [`w @@ f: ModRedElt, ModRedHom → ModRedElt`](morphisms.md#operation-operation-modredelt-modredhom-modredelt-2)

    - [`w @@ f: CombFreeModElt, CombFreeModHom → CombFreeModElt`](morphisms.md#operation-operation-combfreemodelt-combfreemodhom-combfreemodelt-2)