Introduction

 
Construction of Group Algebras and their Elements

      Construction of a Group Algebra
            GroupAlgebra( R, G: parameters ) : Rng, Grp -> AlgGrp
            Example AlgGrp_creation (H93E1)

      Construction of a Group Algebra Element
            elt< A | r, g > : AlgGrp, RngElt, GrpElt -> AlgGrpElt
            A ! g : AlgGrp, GrpElt -> AlgGrpElt
            A ! r : AlgGrp, RngElt -> AlgGrpElt
            A ! [c₁, ..., c_n] : AlgGrp, SeqEnum -> AlgGrpElt
            Eta(A) : AlgGrp -> AlgGrpElt
            Example AlgGrp_el-creation (H93E2)

 
Construction of Subalgebras, Ideals and Quotient Algebras
      sub< cat : A | L> : Cat, AlgGrp, List -> AlgGrp, Map
      lideal< cat : A | L> : Cat, AlgGrp, List -> AlgGrp, Map
      rideal< cat : A | L> : Cat, AlgGrp, List -> AlgGrp, Map
      ideal< cat : A | L> : Cat, AlgGrp, List -> AlgGrp, Map
      a * I : AlgGrpElt, AlgGrpSub -> AlgGrpSub
      I * a : AlgGrpSub, AlgGrpElt -> AlgGrpSub
      quo< A | L > : AlgGrp, List -> AlgAss, Map
      A / S : AlgGrp, AlgGrpSub -> AlgAss
      Example AlgGrp_wedderburn (H93E3)

 
Operations on Group Algebras and their Subalgebras

      Operations on Group Algebras
            Algebra(A) : AlgGrp -> AlgAss, Map
            AugmentationMap(A) : AlgGrp -> Map
            AugmentationIdeal(A) : AlgGrp -> AlgGrpSub
            RepresentationType(A) : AlgGrp -> MonStgElt
            ChangeRepresentationType(A, Rep) : AlgGrp, MonStgElt -> AlgGrp, Map
            ConstructTable(A) : AlgGrp ->
            CoefficientRing(A) : AlgGrp -> Rng

      Operations on Subalgebras of Group Algebras
            S ! 1 : AlgGrpSub -> Grp
            Group(S) : AlgGrpSub -> Grp
            GroupAlgebra(S) : AlgGrpSub -> AlgGrp
            Module(S) : AlgGrpSub -> ModTupRng, Map
            CoefficientRing(A) : AlgGrpSub -> Rng
            BasisMatrix(S) : AlgGrpSub -> ModMatRngElt
            Coordinates(S, a) : AlgGrpSub, AlgGrpElt -> [ RingElt ]
            IsLeftIdeal(S) : AlgGrpSub -> BoolElt
            IsRightIdeal(S) : AlgGrpSub -> BoolElt
            IsIdeal(S) : AlgGrpSub -> BoolElt
            Centraliser(S) : AlgGrpSub -> AlgGrpSub
            Idealiser(S) : AlgGrpSub -> AlgGrpSub
            LeftAnnihilator(S) : AlgGrpSub -> AlgGrpSub
            RightAnnihilator(S) : AlgGrpSub -> AlgGrpSub
            Example AlgGrp_jacobson (H93E4)

 
Operations on Elements
      a + r : AlgGrpElt, RngElt -> AlgGrpElt
      a + g : AlgGrpElt, GrpElt -> AlgGrpElt
      a - r : AlgGrpElt, RngElt -> AlgGrpElt
      a - g : AlgGrpElt, GrpElt -> AlgGrpElt
      a * r : AlgGrpElt, RngElt -> AlgGrpElt
      g * a : GrpElt, AlgGrpElt -> AlgGrpElt
      Support(a) : AlgGrpElt -> SeqEnum
      Trace(a) : AlgGrpElt -> RngElt
      Augmentation(a) : AlgGrpElt -> RngElt
      Involution(a) : AlgGrpElt -> AlgGrpElt
      Coefficient(a, g) : AlgGrpElt, GrpElt -> RngElt
      ElementToSequence(a) : AlgGrpElt -> SeqEnum
      Coefficients(a) : AlgGrpElt -> SeqEnum
      Centraliser(a) : AlgGrpElt -> AlgGrpSub
      Centraliser(S, a) : AlgGrpSub, AlgGrpElt -> AlgGrpSub
      Example AlgGrp_powering (H93E5)
      Example AlgGrp_average (H93E6)

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