Group Representations

• Introduction
  • Definitions and Background
  • Categories
• Creation of Group Representations
  • General Group Representations
    • GroupRepresentation(G, M, action): Grp, CombFreeMod, MonStgElt → ModRed
    • GroupRepresentation(G, M, action): GrpRed, CombFreeMod, MonStgElt → ModRed
  • Subrepresentations
    • Subrepresentation(V, t): ModRed, Any → ModRed, ModRedHom
    • SubCFModule(V, t): CombFreeMod, Any → CombFreeMod, CombFreeModHom
  • Natural Representations
    • TrivialRepresentation(G, R): Grp, Rng → ModRed
    • TrivialRepresentation(G, R): GrpRed, Rng → ModRed
    • StandardRepresentation(G): GrpMat → ModRed
    • StandardRepresentation(G): GrpRed → ModRed
    • SpinorNormRepresentation(G, d): GrpRed, RngIntElt → ModRed
    • Rho(G, k, j): GrpMat, RngIntElt, RngIntElt → ModRed
    • SymSpinor(G, d, k): GrpRed, RngIntElt, RngIntElt → ModRed
    • AltSpinor(G, d): GrpRed, RngIntElt, RngIntElt → ModRed
    • RadicalSignCharacterSinglePrime(G, p): GrpRed, RngIntElt → ModRed
    • RadicalSignCharacter(G, d): GrpRed, RngIntElt → ModRed
    • SpinRepresentation(G, p): GrpRed, RngIntElt → ModRed
  • New Representations from Old
    • DeterminantRepresentation(G): GrpMat → ModRed
    • SymmetricRepresentation(V, n): ModRed, RngIntElt → ModRed
    • AlternatingRepresentation(V, n): ModRed, RngIntElt → ModRed
    • DualRepresentation(V): ModRed → ModRed
    • TensorProduct(V, W): ModRed, ModRed → ModRed
    • TensorPower(V, d): ModRed, RngIntElt → ModRed
    • Pullback(V, f, G): ModRed, MonStgElt, Grp → ModRed
  • New Combinatorially Free Modules from Old
    • ExteriorPower(M, n): CombFreeMod, RngIntElt → CombFreeMod
    • AlternatingPower(M, n): CombFreeMod, RngIntElt → CombFreeMod
    • ExteriorAlgebra(M): CombFreeMod → CombFreeMod, CombFreeModHom
    • DirectSum(M): [ CombFreeMod ] → CombFreeMod
  • Highest Weight Representations
    • GroupRepresentation(G, w): GrpLie, [ RngIntElt ] → ModRed
    • HighestWeightRepresentation(G, w): GrpRed, [ RngIntElt ] → ModRed
    • HighestWeightRepresentation(G, w, p): GrpRed, [ RngIntElt ], RngIntElt → ModRed
  • Creation of Combinatorial Free Modules
    • CombinatorialFreeModule(R, S): Rng, SetIndx → CombFreeMod
    • CombinatorialFreeModule(R, S): Rng, [ MonStgElt ] → CombFreeMod
• Basic Properties
  • Accessing Representation Information
    • Rank(V): ModRed → RngIntElt
    • Dimension(V): ModRed → RngIntElt
    • Ngens(V): ModRed → RngIntElt
    • Basis(V): ModRed → [ ModRedElt ]
    • BaseRing(V): ModRed → Rng
    • Group(V): ModRed → Grp
    • CFM(V): ModRed → CombFreeMod
  • Accessing Combinatorial Free Module Information
    • Rank(M): CombFreeMod → RngIntElt
    • Dimension(M): CombFreeMod → RngIntElt
    • Ngens(M): CombFreeMod → RngIntElt
    • Basis(M): CombFreeMod → [ CombFreeModElt ]
    • BaseRing(M): CombFreeMod → Rng
    • Names(M): CombFreeMod → SetIndx
    • Names(M): CombFreeMod → [ MonStgElt ]
  • Predicates
    • IsTrivial(V): ModRed → BoolElt
• Operations on Group Representations
  • Intersection(V, W): ModRed, ModRed → ModRed
  • V meet W: ModRed, ModRed → ModRed
  • V eq W: ModRed, ModRed → BoolElt
  • V eq W: CombFreeMod, CombFreeMod → BoolElt
  • Base Change
    • ChangeRing(V, S): ModRed, Rng → ModRed
    • ChangeRing(M, S): CombFreeMod, Rng → CombFreeMod
  • Other Operations
    • FixedSubspace(H, V): GrpMat, ModRed → ModRed
• Elements of Group Representations
  • Creation of Elements
    • GroupRepresentationElement(V, m): ModRed, CombFreeModElt → ModRedElt
    • CombinatorialFreeModuleElement(M, v): CombFreeMod, ModRngElt → CombFreeModElt
    • CombinatorialFreeModuleElement(M, v): CombFreeMod, ModEDElt → CombFreeModElt
    • CombinatorialFreeModuleElement(M, v): CombFreeMod, ModTupFldElt[Fld] → CombFreeModElt
  • Basic Properties
    • Parent(v): ModRedElt → ModRed
    • Parent(v): CombFreeModElt → CombFreeMod
    • ActionMatrix(V, g): ModRed, GrpElt → GrpMatElt
  • Operations on Elements
    • v + w: ModRedElt, ModRedElt → ModRedElt
    • v + w: CombFreeModElt, CombFreeModElt → CombFreeModElt
    • v - w: ModRedElt, ModRedElt → ModRedElt
    • v - w: CombFreeModElt, CombFreeModElt → CombFreeModElt
    • a * v: RngElt, ModRedElt → ModRedElt
    • a * v: RngElt, CombFreeModElt → CombFreeModElt
    • g * v: GrpElt, ModRedElt → ModRedElt
    • m * v: AlgMatElt, ModRedElt → ModRedElt
    • v ^ w: CombFreeModElt, CombFreeModElt → CombFreeModElt
  • Comparisons and Membership
    • v eq w: ModRedElt, ModRedElt → BoolElt
    • v in V: ModRedElt, ModRed → BoolElt
    • v in V: CombFreeModElt, CombFreeMod → BoolElt
  • Other Operations
    • Eltseq(v): ModRedElt → []
    • Eltseq(v): CombFreeModElt → []
    • ChangeRing(v, S): CombFreeModElt, Rng → CombFreeModElt
• Homomorphisms of Group Representations
  • Creation of Homomorphisms between Group Representations
    • Homomorphism(V, W, f): ModRed, ModRed, UserProgram → ModRedHom
    • Homomorphism(V, W, f): ModRed, ModRed, Map → ModRedHom
    • Homomorphism(V, W, f): ModRed, ModRed, CombFreeModHom → ModRedHom
    • Homomorphism(V, W, S): ModRed, ModRed, SeqEnum → ModRedHom
    • Homomorphism(M, N, f): CombFreeMod, CombFreeMod, UserProgram → CombFreeModHom
    • Homomorphism(M, N, f): CombFreeMod, CombFreeMod, Map → CombFreeModHom
    • Homomorphism(M, N, S): CombFreeMod, CombFreeMod, SeqEnum → CombFreeModHom
  • Properties of Homomorphisms of Group Representations
    • Domain(f): ModRedHom → ModRed
    • Domain(f): CombFreeModHom → CombFreeMod
    • Codomain(f): ModRedHom → ModRed
    • Codomain(f): CombFreeModHom → CombFreeMod
    • Kernel(f): ModRedHom → ModRed
  • Operations on Homomorphisms of Group Representations
    • Evaluate(f, v): ModRedHom, ModRedElt → ModRedElt
    • Evaluate(f, v): CombFreeModHom, CombFreeModElt → CombFreeModElt
    • v @ f: ModRedElt, ModRedHom → ModRedElt
    • v @ f: CombFreeModElt, CombFreeModHom → CombFreeModElt
    • w @@ f: ModRedElt, ModRedHom → ModRedElt
    • w @@ f: CombFreeModElt, CombFreeModHom → CombFreeModElt