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BettiCharacters > character > character(ActionOnGradedModule,List)

character(ActionOnGradedModule,List) -- compute characters of graded components of a module

Synopsis

Description

This function is provided by the package BettiCharacters.

Use this function to compute the characters of the finite group action on the graded components of a module. The second argument is the (multi)degree of the desired component. For $\mathbb{Z}$-graded rings, one may compute characters in a range of degrees by providing the lowest and highest degrees in the range.

To illustrate, we compute the Betti characters of a symmetric group on the graded components of a polynomial ring, a monomial ideal, and their quotient. The characters are determined by five permutations with cycle types, in order: 4, 31, 22, 211, 1111.

i1 : R = QQ[x_1..x_4]

o1 = R

o1 : PolynomialRing
i2 : I = ideal apply(subsets(gens R,2),product)

o2 = ideal (x x , x x , x x , x x , x x , x x )
             1 2   1 3   2 3   1 4   2 4   3 4

o2 : Ideal of R
i3 : G = {matrix{{x_2,x_3,x_4,x_1}},
          matrix{{x_2,x_3,x_1,x_4}},
          matrix{{x_2,x_1,x_4,x_3}},
          matrix{{x_2,x_1,x_3,x_4}},
          matrix{{x_1,x_2,x_3,x_4}} }

o3 = {| x_2 x_3 x_4 x_1 |, | x_2 x_3 x_1 x_4 |, | x_2 x_1 x_4 x_3 |, | x_2
     ------------------------------------------------------------------------
     x_1 x_3 x_4 |, | x_1 x_2 x_3 x_4 |}

o3 : List
i4 : Q = R/I

o4 = Q

o4 : QuotientRing
i5 : A = action(R,G)

o5 = PolynomialRing with 5 actors

o5 : ActionOnGradedModule
i6 : B = action(I,G)

o6 = Ideal with 5 actors

o6 : ActionOnGradedModule
i7 : C = action(Q,G)

o7 = QuotientRing with 5 actors

o7 : ActionOnGradedModule
i8 : character(A,0,5)

o8 = GradedCharacter{{0} => Character{1, 1, 1, 1, 1}  }
                     {1} => Character{0, 1, 0, 2, 4}
                     {2} => Character{0, 1, 2, 4, 10}
                     {3} => Character{0, 2, 0, 6, 20}
                     {4} => Character{1, 2, 3, 9, 35}
                     {5} => Character{0, 2, 0, 12, 56}

o8 : GradedCharacter
i9 : character(B,0,5)

o9 = GradedCharacter{{0} => Character{0, 0, 0, 0, 0}  }
                     {1} => Character{0, 0, 0, 0, 0}
                     {2} => Character{0, 0, 2, 2, 6}
                     {3} => Character{0, 1, 0, 4, 16}
                     {4} => Character{1, 1, 3, 7, 31}
                     {5} => Character{0, 1, 0, 10, 52}

o9 : GradedCharacter
i10 : character(C,0,5)

o10 = GradedCharacter{{0} => Character{1, 1, 1, 1, 1}}
                      {1} => Character{0, 1, 0, 2, 4}
                      {2} => Character{0, 1, 0, 2, 4}
                      {3} => Character{0, 1, 0, 2, 4}
                      {4} => Character{0, 1, 0, 2, 4}
                      {5} => Character{0, 1, 0, 2, 4}

o10 : GradedCharacter
i11 : character(C,6)

o11 = GradedCharacter{{6} => Character{0, 1, 0, 2, 4}}

o11 : GradedCharacter

See also