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GROUP ALGEBRAS

 
Acknowledgements
 
Introduction
 
Construction of Group Algebras and their Elements
      Construction of a Group Algebra
      Construction of a Group Algebra Element
 
Construction of Subalgebras, Ideals and Quotient Algebras
 
Operations on Group Algebras and their Subalgebras
      Operations on Group Algebras
      Operations on Subalgebras of Group Algebras
 
Operations on Elements







DETAILS

 
Introduction

 
Construction of Group Algebras and their Elements

      Construction of a Group Algebra
            GroupAlgebra( R, G: parameters ) : Rng, Grp -> AlgGrp
            Example AlgGrp_creation (H84E1)

      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, ..., cn] : AlgGrp, SeqEnum -> AlgGrpElt
            Eta(A) : AlgGrp -> AlgGrpElt
            Example AlgGrp_el-creation (H84E2)

 
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 (H84E3)

 
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 (H84E4)

 
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 (H84E5)
      Example AlgGrp_average (H84E6)

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