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character(CharacterDecomposition,CharacterTable) -- recover character from decomposition

Description

Use this function to recover a character from its decomposition into a linear combination of the irreducible characters in a character table. The shortcut d*T is equivalent to the command character(d,T).

As an example, we construct the character table of the symmetric group on 3 elements, then use it to decompose the character of the action of the same symmetric group permuting the variables of a standard graded polynomial ring.

i1 : s = {2,3,1}

o1 = {2, 3, 1}

o1 : List
 
i2 : M = matrix{{1,1,1},{-1,0,2},{1,-1,1}}

o2 = | 1  1  1 |
     | -1 0  2 |
     | 1  -1 1 |

              3       3
o2 : Matrix ZZ  <-- ZZ
 
i3 : P = {1,2,3}

o3 = {1, 2, 3}

o3 : List
 
i4 : T = characterTable(s,M,QQ,P)

o4 = Character table over QQ
      
         |   2   3  1
     ----+-----------
     ꭓ   |   1   1  1
      0  |          
     ꭓ   |  -1   0  2
      1  |          
     ꭓ   |   1  -1  1
      2  |          

o4 : CharacterTable
 
i5 : R = QQ[x_1..x_3]

o5 = R

o5 : PolynomialRing
 
i6 : acts = {matrix{{x_2,x_3,x_1}},matrix{{x_2,x_1,x_3}},matrix{{x_1,x_2,x_3}}}

o6 = {| x_2 x_3 x_1 |, | x_2 x_1 x_3 |, | x_1 x_2 x_3 |}

o6 : List
 
i7 : A = action(R,acts)

o7 = PolynomialRing with 3 actors

o7 : ActionOnGradedModule
 
i8 : c = character(A,0,10)

o8 = Character over QQ
      
      (0, {0})  |  1  1   1
      (0, {1})  |  0  1   3
      (0, {2})  |  0  2   6
      (0, {3})  |  1  2  10
      (0, {4})  |  0  3  15
      (0, {5})  |  0  3  21
      (0, {6})  |  1  4  28
      (0, {7})  |  0  4  36
      (0, {8})  |  0  5  45
      (0, {9})  |  1  5  55
     (0, {10})  |  0  6  66

o8 : Character
 
i9 : d = c/T

o9 = Decomposition table
      
                |  ꭓ   ꭓ   ꭓ
                |   0   1   2
     -----------+------------
      (0, {0})  |   1   0   0
      (0, {1})  |   1   1   0
      (0, {2})  |   2   2   0
      (0, {3})  |   3   3   1
      (0, {4})  |   4   5   1
      (0, {5})  |   5   7   2
      (0, {6})  |   7   9   3
      (0, {7})  |   8  12   4
      (0, {8})  |  10  15   5
      (0, {9})  |  12  18   7
     (0, {10})  |  14  22   8

o9 : CharacterDecomposition
 
i10 : c == d*T

o10 = true

See also

Ways to use this method:

The source of this document is in /build/reproducible-path/macaulay2-1.26.05+ds/M2/Macaulay2/packages/BettiCharacters.m2:3859:0.