Documentation

Introduction
Constructions for A-Modules
- RModule(A) : AlgMat -> ModRng
- RModule(Q) : [ MtrxS ] -> ModTupRng
- Example ModAlg_CreateK6 (H97E1)
Constructions for K[G]-Modules
- General K[G]-Modules
- Natural K[G]-Modules
- Permutation Modules
- Action on an Elementary Abelian Section
  - GModule(G, A, B) : Grp, Grp, Grp -> ModGrp, Map
  - Example ModAlg_CreateA4wrC3 (H97E7)
- Action on a Polynomial Ring
New Modules from Old
- Direct Sums and Tensor Products
- Induction, Restriction and Inflation for K[G]-Modules
- The Fixed-point Spaces for a K[G]-Module
- Change Ring and Base Change
- Writing a K[G]-Module over a Smaller Field
  - Rewriting Over a Smaller Degree Finite Field
  - Rewriting Over a Smaller Degree Number Field
Accessing Module Information
- The Underlying Vector Space
- The Action Algebra
Group Representations
- GModuleAction(M) : ModGrp -> Map(Hom)
- Representation(M) : ModGrp -> Map(Hom)
- Example ModAlg_Representation (H97E12)
- Example ModAlg_Dual (H97E13)
- Kernel(M) : ModGrp -> Grp
- GModuleOfQuotient(M, H) : ModGrp, Grp -> ModGrp
- ActionGenerator(M, i) : ModGrp, RngIntElt -> AlgMatElt
- ActionGenerators(M) : ModGrp -> [ AlgMatElt ]
- NumberOfActionGenerators(M) : ModGrp -> RngIntElt
- ActionGroup(M) : ModGrp -> GrpMat
- Sections(G) : GrpMat -> List
- Example ModAlg_Sections (H97E14)
Module Elements
- Construction
- Deconstruction of Module Elements
  - ElementToSequence(u) : ModRngElt -> [RngElt]
- Action of the Algebra on the Module
  - u * a : ModRngElt, AlgElt -> ModRngElt
  - u * g : ModGrpElt, GrpElt -> ModGrpElt
- Arithmetic with Module Elements
- 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
- Membership and Equality
- Operations on Submodules
  - M + N : ModRng, ModRng -> ModRng
  - M meet N : ModRng, ModRng -> ModRng
- Quotient Modules
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
- Composition Series
- Minimal and Maximal Submodules
- Socle Series
- Decomposition and Complements
Characters and Character Tables
- Ordinary Characters
- Brauer Characters
  - BrauerCharacterTable(G, p) : Chtr -> SeqEnum
  - Character(M) : ModGrp -> AlgChtrElt
Constructing All Irreducible K[G]-Modules
- Constructing Irreducible K[G]-Modules (Characteristic p)
- Constructing Irreducible K[G]-Modules (Characteristic Zero)
- The Cartan Matrix
Lattice of Submodules
Homomorphisms
- Creating Homomorphisms and Hom Spaces
- 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
- The Endomorphism Ring
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
- ProjectiveCover(M) : ModGrp -> ModGrp, ModMatGrpElt
- InjectiveHull(M) : ModGrp -> ModGrp, ModMatGrpElt
- Example ModAlg_Projective Indecomposables (H97E29)
Cohomology and Extensions
- Cohomology
- Extensions of Modules
Vertex and Source of an Indecomposable Module
- Vertex(M : parameters) : ModGrp -> Grp
- Source(M : parameters) : ModGrp -> ModGrp, ModGrp
- Example ModAlg_Vertices and Sources in $M_{22 (H97E33)
Bimodules
- Introduction
- Construction
- Extracting the Left and Right Modules From a Bimodule
- Permutation Bimodules
  - PermutationBimodule(G,H,m,K): Grp, Grp, Map, Fld -> LRModGrp
  - Example ModAlg_permbimodule (H97E37)
- Tensor Products of Bimodules
  - TensorProduct(B1, B2): LRModGrp, LRModGrp -> LRModGrp
  - Example ModAlg_induction (H97E38)
Enumerating All Irreducible Modules
- Irreducible Modules over F_q for Arbitrary Groups
- Irreducible Modules over F_q for Soluble Groups
- Irreducible Modules over Q for Arbitrary Groups
Modules over a General Algebra
- Introduction
- Construction of Algebra Modules
  - Module(A, m): Alg, Map[SetCart, ModRng] -> ModAlg
  - Example ModAlg_AlgModCreate (H97E44)
- The Action of an Algebra Element
- Related Structures of an Algebra Module
- Properties of an Algebra Module
- Creation of Algebra Modules from other Algebra Modules
Bibliography

V2.28, 13 July 2023