Group Algebras

• Introduction
• Construction of Group Algebras and their Elements
  • Construction of a Group Algebra
    • GroupAlgebra( R, G: parameters ): Rng, Grp → AlgGrp
    • GroupAlgebra< R, G: parameters >: Rng, Grp → AlgGrp
    • Example: Constructing a Group Algebra
  • 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: Idempotent in a Group Algebra
• 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: Wedderburn Decomposition of a Group Algebra
• 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
    • BaseRing(A): AlgGrp → Rng
  • Operations on Subalgebras of Group Algebras
    • S ! 1: AlgGrpSub → Grp
    • S ! 1: AlgGrp → Grp
    • Group(S): AlgGrpSub → Grp
    • Group(S): AlgGrp → Grp
    • GroupAlgebra(S): AlgGrpSub → AlgGrp
    • GroupAlgebra(S): AlgGrp → AlgGrp
    • Module(S): AlgGrpSub → ModTupRng, Map
    • Module(S): AlgGrp → ModTupRng, Map
    • CoefficientRing(A): AlgGrpSub → Rng
    • BaseRing(A): AlgGrpSub → Rng
    • BasisMatrix(S): AlgGrpSub → ModMatRngElt
    • BasisMatrix(S): AlgGrp → ModMatRngElt
    • Coordinates(S, a): AlgGrpSub, AlgGrpElt → [ RingElt ]
    • Coordinates(S, a): AlgGrp, AlgGrpElt → [ RingElt ]
    • IsLeftIdeal(S): AlgGrpSub → BoolElt
    • IsLeftIdeal(S): AlgGrp → BoolElt
    • IsRightIdeal(S): AlgGrpSub → BoolElt
    • IsRightIdeal(S): AlgGrp → BoolElt
    • IsIdeal(S): AlgGrpSub → BoolElt
    • IsIdeal(S): AlgGrp → BoolElt
    • Centraliser(S): AlgGrpSub → AlgGrpSub
    • Centralizer(S): AlgGrpSub → AlgGrpSub
    • Centraliser(S): AlgGrp → AlgGrpSub
    • Centralizer(S): AlgGrp → AlgGrpSub
    • Idealiser(S): AlgGrpSub → AlgGrpSub
    • Idealizer(S): AlgGrpSub → AlgGrpSub
    • Idealiser(S): AlgGrp → AlgGrpSub
    • Idealizer(S): AlgGrp → AlgGrpSub
    • LeftAnnihilator(S): AlgGrpSub → AlgGrpSub
    • LeftAnnihilator(S): AlgGrp → AlgGrpSub
    • RightAnnihilator(S): AlgGrpSub → AlgGrpSub
    • RightAnnihilator(S): AlgGrp → AlgGrpSub
    • Example: Jacobson Radical of a Group Algebra
• Operations on Elements
  • a + r: AlgGrpElt, RngElt → AlgGrpElt
  • r + a: RngElt, AlgGrpElt → AlgGrpElt
  • a + g: AlgGrpElt, GrpElt → AlgGrpElt
  • g + a: GrpElt, AlgGrpElt → AlgGrpElt
  • a - r: AlgGrpElt, RngElt → AlgGrpElt
  • r - a: RngElt, AlgGrpElt → AlgGrpElt
  • a - g: AlgGrpElt, GrpElt → AlgGrpElt
  • g - a: GrpElt, AlgGrpElt → AlgGrpElt
  • a * r: AlgGrpElt, RngElt → AlgGrpElt
  • r * a: RngElt, AlgGrpElt → AlgGrpElt
  • g * a: GrpElt, AlgGrpElt → AlgGrpElt
  • a * g: AlgGrpElt, GrpElt → AlgGrpElt
  • Support(a): AlgGrpElt → SeqEnum
  • Trace(a): AlgGrpElt → RngElt
  • Augmentation(a): AlgGrpElt → RngElt
  • Involution(a): AlgGrpElt → AlgGrpElt
  • Coefficient(a, g): AlgGrpElt, GrpElt → RngElt
  • a[g]: AlgGrpElt, GrpElt → RngElt
  • ElementToSequence(a): AlgGrpElt → SeqEnum
  • Eltseq(a): AlgGrpElt → SeqEnum
  • Coefficients(a): AlgGrpElt → SeqEnum
  • Centraliser(a): AlgGrpElt → AlgGrpSub
  • Centralizer(a): AlgGrpElt → AlgGrpSub
  • Centraliser(S, a): AlgGrpSub, AlgGrpElt → AlgGrpSub
  • Centralizer(S, a): AlgGrpSub, AlgGrpElt → AlgGrpSub
  • Example: Diameter of a Cayley Graph
  • Example: Random Distribution of Words