Modules over an Algebra and Group Representations

• Introduction
• Constructions for \(A\)-Modules
  • RModule(A): AlgMat → ModRng
  • RModule(Q): [ MtrxS ] → ModTupRng
  • Example: Module of Dimension 6 over a Finite Field
• Constructions for \(K[G]\)-Modules
  • General \(K[G]\)-Modules
    • GModule(G, A): Grp, AlgMat → ModGrp
    • GModule(G, Q): Grp, [ GrpMatElt ] → ModGrp
    • TrivialModule(G, K): Grp, Fld → ModGrp
    • Example: Three-Dimensional Module for PSL(2,7)
    • Example: Module from a Conjugation Action
  • Natural \(K[G]\)-Modules
    • GModule(G, K): GrpPerm, Rng → ModGrp
    • GModule(G): GrpMat → ModGrp
    • Example: Natural Module for M11
  • Permutation Modules
    • PermutationModule(G, H, K): Grp, Grp, Fld → ModGrp
    • PermutationModule(G, K): Grp, Fld → ModGrp
    • PermutationModule(G, V): Grp, ModTupFld → ModGrp
    • PermutationModule(G, u): Grp, ModTupFldElt → ModGrp
    • Example: Permutation Module for M12
    • Example: Constituent of a Permutation Module for A7
  • Action on an Elementary Abelian Section
    • GModule(G, A, B): Grp, Grp, Grp → ModGrp, Map
    • GModule(G, A): Grp, Grp → ModGrp, Map
    • Example: Module for the Wreath Product A4 wr C3
  • Action on a Polynomial Ring
    • GModule(G, P, d): Grp, RngMPol, RngIntElt → ModGrp, Map, {@ RngMPolElt @}
    • GModule(G, I, J): Grp, RngMPol, RngMPol → ModGrp, Map, {@ RngMPolElt @}
    • GModule(G, Q): Grp, RngMPolRes → ModGrp, Map, {@ RngMPolElt @}
    • Example: Representation on Homogeneous Polynomials
• New Modules from Old
  • Direct Sums and Tensor Products
    • DirectSum(M, N): ModRng, ModRng → ModRng, Map, Map, Map, Map
    • DirectSum(Q): [ ModRng ] → ModRng, [ Map ], [ Map ]
    • TensorProduct(M, N): ModMat, ModMat → ModMat
    • TensorProduct(M, N): ModGrp, ModGrp → ModGrp
    • TensorPower(M, n): ModMat, RngIntElt → ModMat
    • ExteriorSquare(M): ModMat → ModMat
    • SymmetricSquare(M): ModMat → ModMat
    • GTensorProduct(M, N): ModGrp, ModGrp → ModGrp, Map
    • GTensorProduct(M, N, H): ModGrp, ModGrp, Grp → ModGrp, Map
  • Induction, Restriction, and Inflation for \(K[G]\)-Modules
    • Dual(M): ModGrp → ModGrp
    • Induction(M, G): ModGrp, Grp → ModGrp
    • Induction(R, G): Map, Grp → Map
    • Restriction(M, H): ModGrp, Grp → ModGrp
    • Inflation(M, h): ModGrp, Map → ModGrp
    • Example: Induction and Restriction for M22
  • The Fixed-point Spaces for a \(K[G]\)-Module
    • Fix(M): Mod → Mod
    • FixMod(M, H): ModGrp, Grp → Mod
    • FixDual(M): ModGrp → Modgrp, Map
    • FixDualMod(M,H): ModGrp, Grp → Mod
  • Change Ring and Base Change
    • ChangeRing(M, S): ModRng, Rng → ModRng, Map
    • ChangeRing(M, S, f): ModRng, Rng, Map → ModRng, Map
    • M ^ T: ModGrp, AlgMatElt → ModGrp
  • Writing a \(K[G]\)-Module over a Smaller Field
    • Rewriting Over a Smaller Degree Finite Field
      • IsRealisableOverSmallerField(M): ModGrp → BoolElt, ModGrp
      • IsRealisableOverSubfield(M, F): ModGrp, FldFin → BoolElt, ModGrp
      • WriteOverSmallerField(M, F): ModGrp, FldFin → ModGrp, Map
      • AbsoluteModuleOverMinimalField(M, F): ModGrp, FldFin → ModGrp
      • AbsoluteModuleOverMinimalField(M): ModGrp → ModGrp
      • AbsoluteModulesOverMinimalField(Q, F): [ ModGrp ], FldFin → [ ModGrp ]
      • ModuleOverSmallerField(M, F): ModGrp, FldFin → ModGrp
      • ModulesOverSmallerField(Q, F): SeqEnum, FldFin → ModGrp
      • ModulesOverCommonField(M, N): ModGrp, ModGrp → ModGrp, ModGrp
    • Rewriting Over a Smaller Degree Number Field
      • AbsoluteModuleOverMinimalField(M): ModGrp → ModGrp
      • Minimize(M): ModGrp → ModGrp
      • Minimize(R): Map → Map
      • WriteGModuleOver(M, K): ModGrp, FldAlg → ModGrp
      • WriteRepresentationOver(R, K): Map, FldAlg → Map
      • Example: Minimal Field of a Q8 Representation
• Accessing Module Information
  • The Underlying Vector Space
    • M . i: ModRng, RngIntElt → ModElt
    • CoefficientRing(M): ModRng → Rng
    • BaseRing(M): ModRng → Rng
    • Generators(M): ModRng → { ModRngElt }
    • Parent(u): ModRngElt → ModRng
  • The Action Algebra
    • Action(M): ModRng → AlgMat
    • RightAction(M): ModRng → AlgMat
    • MatrixGroup(M): ModGrp → GrpMat
    • ActionGenerator(M, i): ModRng, RngIntElt → AlgMatElt
    • NumberOfActionGenerators(M): ModRng → RngIntElt
    • Ngens(M): ModTupRng → RngIntElt
    • Group(M): ModGrp → Grp
    • Example: Access Functions for a Matrix Algebra Module
• Group Representations
  • GModuleAction(M): ModGrp → Map(Hom)
  • Representation(M): ModGrp → Map(Hom)
  • Example: Characters from a Representation of A7
  • Example: Constructing the Dual Module
  • Kernel(M): ModGrp → Grp
  • GModuleOfQuotient(M, H): ModGrp, Grp → ModGrp
  • ActionGenerator(M, i): ModGrp, RngIntElt → AlgMatElt
  • RightActionGenerator(M, i): ModGrp, RngIntElt → AlgMatElt
  • ActionGenerators(M): ModGrp → [ AlgMatElt ]
  • NumberOfActionGenerators(M): ModGrp → RngIntElt
  • Nagens(M): ModGrp → RngIntElt
  • ActionGroup(M): ModGrp → GrpMat
  • Sections(G): GrpMat → List
  • Example: Composition Factors of a Tensor Square
• Module Elements
  • Construction
    • elt< M | a₁, ..., aₙ >: ModRng, List → ModRngElt
    • M ! Q: ModRng, [RngElt] → ModRngElt
    • Zero(M): ModRng → ModRngElt
    • M ! 0: ModRng, RngIntElt → ModRngElt
    • Random(M): ModRng → ModRngElt
  • Deconstruction of Module Elements
    • ElementToSequence(u): ModRngElt → [RngElt]
    • Eltseq(u): ModRngElt → [RngElt]
  • Action of the Algebra on the Module
    • u * a: ModRngElt, AlgElt → ModRngElt
    • u * g: ModGrpElt, GrpElt → ModGrpElt
  • Arithmetic with Module Elements
    • u + v: ModRngElt, ModRngElt → ModRngElt
    • - u: ModRngElt → ModRngElt
    • u - v: ModRngElt, ModRngElt → ModRngElt
    • k * u: RngElt, ModRngElt → ModRngElt
    • u * k: ModRngElt, RngElt → ModRngElt
    • u / k: ModRngElt, RngElt → ModRngElt
  • Indexing
    • u[i]: ModRngElt, RngIntElt → RngElt
    • u[i] := x: ModRngElt, RngIntElt, RngElt → ModRngElt
  • Properties of Module Elements
    • IsZero(u): ModRngElt → BoolElt
    • Support(u): ModRngElt → { RngIntElt }
• Submodules and Quotient Modules
  • Construction
    • sub<M | L>: ModRng, List → ModRng
    • ImageWithBasis(X, M): ModMatRngElt, ModRng → ModRng
    • Morphism(M, N): ModRng, ModRng → ModMatRngElt
    • Example: Submodule of a Permutation Module for L(3,4)
  • Membership and Equality
    • u in M: ModRngElt, ModRng → BoolElt
    • N subset M: ModRng, ModRng → BoolElt
    • N eq M: ModRng, ModRng → BoolElt
  • Operations on Submodules
    • M + N: ModRng, ModRng → ModRng
    • M meet N: ModRng, ModRng → ModRng
  • Quotient Modules
    • quo<M | L>: ModRng, List → ModRng
    • Morphism(M, N): ModRng, ModRng → ModMatRngElt
    • Example: Quotient of a Permutation Module for L(3,4)
• Properties of a Module
  • IsIrreducible(M): ModRng → BoolElt, ModRng, ModRng
  • IsAbsolutelyIrreducible(M): ModRng → BoolElt, AlgMatElt, RngIntElt
  • IsDecomposable(M): ModRng → BoolElt, ModRng, ModRng
  • IsSemisimple(M): ModGrp → BoolElt
  • IsProjective(M): ModGrp → BoolElt
  • IsFree(M): ModGrp → BoolElt
  • IsSelfDual(M): ModGrp → BoolElt
  • IsPermutationModule(M): ModRng → BoolElt
• Structure of a Module
  • Splitting a Module
    • Meataxe(M): ModRng → ModRng, ModRng, AlgMatElt
    • IsIrreducible(M): ModRng → BoolElt, ModRng, ModRng
    • IsAbsolutelyIrreducible(M): ModRng → BoolElt, AlgMatElt, RngIntElt
    • AbsolutelyIrreducibleModule(M): ModRng → ModRng
    • MinimalField(M): ModRng → FldFin
    • Example: Applying the Meataxe to O5(3)
  • Composition Series
    • CompositionSeries(M): ModRng → [ ModRng ], [ ModRng ], AlgMatElt
    • CompositionFactors(M): ModRng → [ ModRng ]
    • Constituents(M): ModRng → [ ModRng ], [ RngIntElt ]
    • ConstituentsWithMultiplicities(M): ModRng → [ <ModRng, RngIntElt> ], [ RngIntElt ]
    • Example: Composition Series for O5(3)
  • Minimal and Maximal Submodules
    • JacobsonRadical(M): ModRng → ModRng, Map
    • MinimalSubmodule(M): ModRng → ModRng
    • MinimalSubmodules(M): ModRng → [ ModRng ], BoolElt
    • MinimalSubmodules(M, F): ModRng, ModRng → [ ModRng ], BoolElt
    • MaximalSubmodules(M): ModRng → [ ModRng ], BoolElt
  • Socle Series
    • Socle(M): ModRng → ModRng, Map
    • SocleRecursive(M): ModRng → ModRng
    • SocleSeries(M): ModRng → [ ModRng ], [ ModRng ], AlgMatElt
    • SocleFactors(M): ModRng → [ ModRng ]
    • SocleLayerFactors(M): ModGrp → SeqEnum
    • DisplaySocleStructure(SL): SeqEnum
    • Example: Minimal Submodules for O5(3)
  • Decomposition and Complements
    • IsDecomposable(M): ModRng → BoolElt, ModRng, ModRng
    • IsSemisimple(M): ModGrp → BoolElt
    • DirectSumDecomposition(M): ModRng → [ ModRng ]
    • IndecomposableSummands(M): ModRng → [ ModRng ]
    • Decomposition(M): ModRng → [ ModRng ]
    • RelativeDecomposition(M, T): ModRng, ModRng → ModRng, ModRng
    • HasComplement(M, S): ModGrp, ModGrp → BoolElt, ModGrp
    • HasComplement(M, S): ModRng, ModRng → BoolElt, ModRng
    • IsDirectSummand(M, S): ModGrp, ModGrp → BoolElt, ModGrp
    • IsDirectSummand(M, S): ModRng, ModRng → BoolElt, ModRng
    • Complements(M, S): ModGrp, ModGrp → [ ModGrp ]
    • Complements(M, S): ModRng, ModRng → [ ModRng ]
    • Example: Decomposition of a Matrix Algebra Module
• Characters and Character Tables
  • Ordinary Characters
    • Character(M): ModGrp → AlgChtrElt
    • CharacterTable(G): Chtr → SeqEnum
    • SymmetricCharacterTable(n): RngIntElt → SeqEnum
    • AlternatingCharacterTable(n): RngIntElt → SeqEnum
    • RationalCharacterTable(G): Chtr → SeqEnum
  • Brauer Characters
    • BrauerCharacterTable(G, p): Chtr → SeqEnum
    • Character(M): ModGrp → AlgChtrElt
• Constructing All Irreducible \(K[G]\)-Modules
  • Constructing Irreducible \(K[G]\)-Modules (Characteristic \(p\))
    • IrreducibleModules(G, K : parameters): Grp, Fld → SeqEnum
    • AbsolutelyIrreducibleModules(G, K : parameters): Grp, Fld → SeqEnum
    • AbsolutelyIrreducibleModulesSoluble(G, p): Grp, RngIntElt → SeqEnum
    • AbsolutelyIrreducibleModulesSchur(G, K): GrpPC, Fld → List
    • Example: Irreducible Modules for PSL(3,4) in Characteristic 3
  • Constructing Irreducible \(K[G]\)-Modules (Characteristic Zero)
    • GModule(x): Chtr → ModGrp
    • IrreducibleModules(G, K): Grp, Fld → SeqEnum
    • AbsolutelyIrreducibleModules(G): Grp → SeqEnum
    • AbsolutelyIrreducibleModulesSchur(G, K): GrpPC, Fld → List
    • Example: Irreducible Modules for PSL(3,4) over the Rationals
  • The Cartan Matrix
    • CartanMatrix(G, K): Grp, FldFin → AlgMatElt
    • AbsoluteCartanMatrix(G, K): Grp, FldFin → AlgMatElt
    • DecompositionMatrix(G, K): Grp, FldFin → AlgMatElt
• Lattice of Submodules
  • Creating Lattices
    • SubmoduleLattice(M): ModRng → SubModLat, BoolElt
    • SubmoduleLatticeAbort(M, n): ModRng, RngIntElt → BoolElt, SubModLat
    • SetVerbose("SubmoduleLattice", i): MonStgElt, RngIntElt
    • Submodules(M): ModRng → [ModRng]
    • Example: Creating a Submodule Lattice
  • Operations on Lattices
    • # L: SubModLat → RngIntElt
    • L ! i: SubModLat, RngIntElt → SubModLatElt
    • L ! S: SubModLat, ModRng → SubModLatElt
    • Bottom(L): SubModLat → SubModLatElt
    • Random(L): SubModLat → SubModLatElt
    • Top(L): SubModLat → SubModLatElt
  • Operations on Lattice Elements
    • IntegerRing() ! e: RngInt, SubModLatElt → RngIntElt
    • e + f: SubModLatElt, SubModLatElt → SubModLatElt
    • e meet f: SubModLatElt, SubModLatElt → SubModLatElt
    • e eq f: SubModLatElt, SubModLatElt → SubModLatElt
    • e subset f: SubModLatElt, SubModLatElt → SubModLatElt
    • MaximalSubmodules(e): SubModLatElt → { SubModLatElt }
    • MinimalSupermodules(e): SubModLatElt → { SubModLatElt }
    • Module(e): SubModLatElt → ModRng
  • Properties of Lattice Elements
    • Dimension(e): SubModLatElt → RngIntElt
    • JacobsonRadical(e): SubModLatElt → SubModLatElt
    • Morphism(e): SubModLatElt → ModMatRngElt
    • Example: Operations in a Submodule Lattice
• Homomorphisms
  • Creating Homomorphisms and Hom Spaces
    • hom< M -> N | X >: ModRng, ModRng, ModMatElt → Map
    • Hom(M, N): ModRng, ModRng → ModMatRng
    • GHom(M, N): ModGrp, ModGrp → ModMatGrp
    • GHomOverCentralizingField(M, N): ModGrp, ModGrp → ModMatGrp
    • GHomOverCentralizingField(M, N): ModRng, ModRng → ModRng
    • AHom(M, N): ModRng, ModRng → ModMatRng
    • HomMod(M, N): ModGrp, ModGrp → ModGrp
    • H ! f: ModMatRng, Map → ModMatRngElt
    • IsModuleHomomorphism(X): ModMatFldElt → BoolElt
    • Example: Hom Space of Modules for a Soluble Group
    • Example: Endomorphisms of a G-Homomorphism Module
  • Isomorphism and Similarity
    • Isomorphism
      • IsIsomorphic(M, N): ModRng, ModRng → ModRng, ModRng, BoolElt, AlgMatElt
      • SummandIsomorphism(M, N): ModRng, ModRng → ModRng, ModRng, Map, Map
    • Similarity of Cyclic Algebras and their Modules
      • IsCyclic(R): AlgAss → BoolElt, AlgAssElt
      • IsSimilar(M, N): ModRng, ModRng → BoolElt, Map
      • Example: Similarity versus Isomorphism of Modules
  • The Endomorphism Ring
    • EndomorphismAlgebra(M): ModRng → AlgMat
    • EndomorphismRing(M): ModRng → AlgMat
    • CentreOfEndomorphismRing(M): ModRng → AlgMat
    • AutomorphismGroup(M): ModRng → GrpMat
    • Example: Endomorphism Ring of a Permutation Module
• Projective Indecomposable Modules
  • ProjectiveIndecomposableDimensions(G, K): Grp, FldFin → SeqEnum
  • ProjectiveIndecomposableModule(I: parameters): ModGrp → ModGrp
  • ProjectiveIndecomposableModules(G, K: parameters): Grp, FldFin → SeqEnum
  • PIMBlocks(CM): AlgMatElt → SeqEnum
  • PIMBlocks(I, P): SeqEnum, SeqEnum → SeqEnum
  • ProjectiveCover(M): ModGrp → ModGrp, ModMatGrpElt
  • InjectiveHull(M): ModGrp → ModGrp, ModMatGrpElt
  • Example: Projective Indecomposables for Alt(8)
• Cohomology and Extensions
  • Cohomology
    • CohomologyModule(G, M): GrpPerm, ModGrp → ModCoho
    • CohomologyModule(G, M): GrpPC, ModGrp → ModCoho
    • CohomologyGroup(CM, n): ModCoho, RngIntElt → ModTupRng
    • CohomologicalDimension(CM, n): ModCoho, RngIntElt → RngIntElt
    • CohomologicalDimension(M, n): ModGrp, RngIntElt → RngIntElt
    • CohomologicalDimensions(M, n): ModGrp, RngIntElt → RngIntElt
    • Example: Cohomology of a PSL(3,3) Module
    • Example: Cohomological Dimension over Alt(8)
  • Extensions of Modules
    • Ext(M, N): ModGrp, ModGrp → ModTupFld
    • Extension(M, N, e, r): ModGrp, ModGrp, ModTupFldElt, Map → ModGrp, ModMatGrpElt, ModMatGrpElt
    • MaximalExtension(M, N, E, r): ModGrp, ModGrp, ModTupFld, Map → ModGrp
    • MaximalExtension(M, N): ModGrp, ModGrp → ModGrp
    • MaximalExtension(~M, N): ModGrp, ModGrp
    • Example: Extensions of Alt(5) Modules
    • LowDimensionalModules(G, K, n): Grp, Fld, RngIntElt → SeqEnum
• Vertex and Source of an Indecomposable Module
  • Vertex(M : parameters): ModGrp → Grp
  • Source(M : parameters): ModGrp → ModGrp, ModGrp
  • Example: Vertices and Sources in M22.2
• Bimodules
  • Introduction
  • Construction
    • Bimodule(M,N): ModGrp, ModGrp → LRModGrp
    • Example: Bimodule from Sym(3) and a Cyclic Group
    • Bimodule(G,H,M): Grp, Grp, ModGrp → LRModGrp
    • Example: Bimodule from a Direct Product
    • LeftBimodule(M): ModGrp → LRModGrp
    • RightBimodule(M): ModGrp → LRModGrp
  • Extracting the Left and Right Modules From a Bimodule
    • LeftOppositeModule(B): LRModGrp → ModGrp
    • RightModule(B): LRModGrp → ModGrp
    • Example: Left and Right Modules of a Bimodule
  • Permutation Bimodules
    • PermutationBimodule(G,H,m,K): Grp, Grp, Map, Fld → LRModGrp
    • Example: Permutation Bimodules
  • Tensor Products of Bimodules
    • TensorProduct(B1, B2): LRModGrp, LRModGrp → LRModGrp
    • Example: Induced Modules via Tensor Products
• Enumerating All Irreducible Modules
  • Irreducible Modules over \(F_q\) for Arbitrary Groups
    • IrreducibleModules(G, K : parameters): Grp, Fld → SeqEnum
    • AbsolutelyIrreducibleModules(G, K : parameters): Grp, Fld → SeqEnum
    • Example: Irreducible Modules of a Group of Order 416
    • AbsolutelyIrreducibleModulesBurnside(G, K : parameters ): Grp, FldFin → [ ModGrp ]
    • IrreducibleModulesBurnside(G, K : parameters ): Grp, FldFin → [ ModGrp ]
    • AbsolutelyIrreducibleConstituents(M): ModGrp → [ ModGrp ]
    • Example: Irreducible Modules for M11
  • Irreducible Modules over \(F_q\) for Soluble Groups
    • IrreducibleModules(G, K : parameters): Grp, Fld → SeqEnum
    • AbsolutelyIrreducibleModulesSchur(G, K: parameters): GrpPC, Rng → List[GMod ule]
    • AbsolutelyIrreducibleModulesSchur(G, k, i: parameters): GrpPC, Rng, RngIntElt → List[GModule]
    • AbsolutelyIrreducibleModulesSchur(G, k, L: parameters): GrpPC, Rng, List[GModule] → List[GModule]
    • AbsolutelyIrreducibleModulesSchur(G, k, L, i: parameters): GrpPC, Rng, RngIntElt → List[GModule]
    • IrreducibleModulesSchur(G, K: parameters): GrpPC, Rng → List[GModule]
    • IrreducibleModulesSchur(G, K, i: parameters): GrpPC, Rng, RngIntElt → List[GModule]
    • IrreducibleModulesSchur(G, K, L: parameters): GrpPC, Rng, List[GModule] → List[GModule]
    • IrreducibleModulesSchur(G, K, L, i: parameters): GrpPC, Rng, List[GModule], RngIntElt → List[GModule]
    • Example: Representations of a Dihedral Group of Order 20
  • Irreducible Modules over \({\mathbb{Q}}\) for Arbitrary Groups
    • IrreducibleModules(G, Q : parameters): Grp, FldRat → SeqEnum, SeqEnum
    • RationalCharacterTable(G): Grp → SeqEnum, SeqEnum
    • Example: Rational Irreducible Modules for PSL(3,3)
    • Example: Irreducible Rational Modules for M23
• Modules over a General Algebra
  • Introduction
  • Construction of Algebra Modules
    • Module(A, m): Alg, Map[SetCart, ModRng] → ModAlg
    • Example: Natural Module of a Matrix Algebra
  • The Action of an Algebra Element
    • a ^ v: AlgElt, ModAlgElt → ModAlgElt
    • v ^ a: ModAlgElt, AlgElt → ModAlgElt
    • ActionMatrix(M, a): ModAlg, AlgElt → AlgMatElt
    • Example: Action Matrices of Algebra Elements
  • Related Structures of an Algebra Module
    • Algebra(M): ModAlg → Alg
    • CoefficientRing(M): ModAlg → Fld
    • Basis(M): ModAlg → SeqEnum
  • Properties of an Algebra Module
    • IsLeftModule(M): ModAlg → BoolElt
    • IsRightModule(M): ModAlg → BoolElt
    • Dimension(M): ModAlg → RngIntElt
  • Creation of Algebra Modules from other Algebra Modules
    • DirectSum(Q): SeqEnum → ModAlg, SeqEnum, SeqEnum
    • SubalgebraModule(B, M): Alg, ModAlg → ModAlg
    • ModuleWithBasis(Q): SeqEnum → ModAlg
    • Example: Direct Sums of Algebra Modules
    • sub< M | S >: ModAlg, [ModAlgElt] → ModAlg
    • sub< M | e1, ..., en >: ModAlg, ModAlgElt, ..., ModAlgElt → ModAlg
    • quo< M | S >: ModAlg, [ModAlgElt] → ModAlg
    • quo< M | e1, ..., en >: ModAlg, ModAlgElt, ..., ModAlgElt → ModAlg
    • quo< M | S >: ModAlg, ModAlg → ModAlg
• Bibliography