BettiCharacters : Table of Contents

• BettiCharacters -- finite group characters on free resolutions and graded modules
  • action -- define finite group action
    • Action -- the class of all finite group actions
      • ActionOnComplex -- the class of all finite group actions on complexes
      • ActionOnGradedModule -- the class of all finite group actions on graded modules
      • ringActors -- get action on ring generators
      • net(Action) -- format for printing, as a net
    • action(Complex,List,List,ZZ) -- define finite group action on a resolution
    • action(Module,List,List) -- define finite group action on a graded module
    • Semidirect -- action of semidirect product with torus
    • Sub -- format ring actors as one-row substitution matrices
  • actors -- group elements of an action
    • actors(ActionOnComplex,ZZ) -- group elements of action on resolution
    • actors(ActionOnGradedModule,List) -- group elements acting on components of a module
  • Character class -- internal representation of graded characters
  • character -- compute characters of finite group action
    • Character -- the class of all characters of finite group representations
      • Character Array -- homological shift
      • RingElement * Character -- scalar multiple of a character
      • Character ^ List -- extract graded component
      • Character _ ZZ -- extract component
      • minus(Character) -- additive inverse of a character
      • directSum(Character) -- direct sum of characters
      • difference(Character,Character) -- difference of characters
      • dual -- dual character
      • net(Character) -- format for printing, as a net
      • tensor(Character,Character) -- tensor product of characters
      • texMath(Character) -- convert to TeX math format
    • character(ActionOnComplex) -- compute all Betti characters of minimal free resolution
    • character(ActionOnComplex,ZZ) -- compute Betti characters of minimal free resolution
    • character(ActionOnGradedModule,List) -- compute characters of graded components of a module
    • character(PolynomialRing,HashTable) -- construct a character
    • character(CharacterDecomposition,CharacterTable) -- recover character from decomposition
  • Character operations -- including shift, direct sum, dual, and tensor product
  • characterTable -- construct a character table
    • CharacterTable -- the class of all character tables of finite groups
      • net(CharacterTable) -- format for printing, as a net
      • texMath(CharacterTable) -- convert to TeX math format
    • Labels -- custom labels for irreducible characters
  • decomposeCharacter -- decompose a character into irreducible characters
    • CharacterDecomposition -- the class of all finite group character decompositions
      • net(CharacterDecomposition) -- format for printing, as a net
      • texMath(CharacterDecomposition) -- convert to TeX math format
  • Equality checks -- compare actions and characters
  • symmetricGroupActors -- permutation action of the symmetric group
  • symmetricGroupTable -- character table of the symmetric group
  • hyperoctahedralGroupActors -- standard action of the hyperoctahedral group
  • hyperoctahedralGroupTable -- character table of the hyperoctahedral group
  • BettiCharacters Example 1 -- Specht ideals / subspace arrangements
  • BettiCharacters Example 2 -- Symbolic powers of star configurations
  • BettiCharacters Example 3 -- Klein configuration of points
  • BettiCharacters Example 4 -- a multigraded example
  • BettiCharacters Example 5 -- semidirect product of torus and symmetric group