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
      • net(Action) -- format for printing, as a net
      • ring(Action) -- get ring of object acted upon
      • ringActors -- get action on ring generators
      • target(Action) -- get object acted upon
    • action(ChainComplex,List,List,ZZ) -- define finite group action on a resolution
    • action(Module,List,List) -- define finite group action on a graded module
    • 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
    • inverseRingActors -- get inverse of action on ring generators
    • numActors -- number of acting elements
  • character -- compute characters of finite group action

    • Character -- the class of all characters of finite group representations

      • Character Array -- homological shift
      • directSum(Character) -- direct sum of characters
      • dual -- dual character
      • net(Character) -- format for printing, as a net
      • tensor(Character,Character) -- tensor product of characters
    • 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,ZZ,HashTable) -- construct a character
    • character(CharacterDecomposition,CharacterTable) -- recover character from decomposition
  • Character operations -- 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
    • 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
  • symmetricGroupActors -- permutation action of the symmetric group
  • symmetricGroupTable -- character table of the symmetric group
  • BettiCharacters Example 1 -- Specht ideals / subspace arrangements
  • BettiCharacters Example 2 -- Symbolic powers of star configurations
  • BettiCharacters Example 3 -- Klein configuration of points