Introduction

      Definitions and Background

      Categories

 
Creation of Group Representations

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

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

      Natural Representations
            TrivialRepresentation(G, R) : Grp, Rng -> ModRed
            StandardRepresentation(G) : GrpMat -> 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
            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

 
Basic Properties

      Accessing Representation Information
            Rank(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
            Basis(M) : CombFreeMod -> [ CombFreeModElt ]
            BaseRing(M) : CombFreeMod -> Rng
            Names(M) : CombFreeMod -> SetIndx

      Predicates
            IsTrivial(V) : ModRed -> BoolElt

 
Operations on Group Representations
      Intersection(V, W) : ModRed, ModRed -> ModRed
      V eq W : ModRed, ModRed -> 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

      Basic Properties
            Parent(v) : ModRedElt -> ModRed
            ActionMatrix(V, g) : ModRed, GrpElt -> GrpMatElt

      Operations on Elements
            v + w : ModRedElt, ModRedElt -> ModRedElt
            v - w : ModRedElt, ModRedElt -> ModRedElt
            a * v : RngElt, ModRedElt -> ModRedElt
            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

      Other Operations
            Eltseq(v) : ModRedElt -> []
            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, S) : ModRed, ModRed, SeqEnum -> ModRedHom
            Homomorphism(M, N, f) : CombFreeMod, CombFreeMod, UserProgram -> CombFreeModHom
            Homomorphism(M, N, S) : CombFreeMod, CombFreeMod, SeqEnum -> CombFreeModHom

      Properties of Homomorphisms of Group Representations
            Domain(f) : ModRedHom -> ModRed
            Codomain(f) : ModRedHom -> ModRed
            Kernel(f) : ModRedHom -> ModRed

      Operations on Homomorphisms of Group Representations
            Evaluate(f, v) : ModRedHom, ModRedElt -> ModRedElt
            w @@ f : ModRedElt, ModRedHom -> ModRedElt

 
Bibliography

[Next][Prev] [____] [Up] [Index] [Root]