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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 : R = QQ[x_1..x_3]

o3 = R

o3 : PolynomialRing
 
i4 : P = {1,2,3}

o4 = {1, 2, 3}

o4 : List
 
i5 : T = characterTable(s,M,R,P)

o5 = Character table over R
      
         |   2   3  1
     ----+-----------
     X0  |   1   1  1
     X1  |  -1   0  2
     X2  |   1  -1  1

o5 : CharacterTable
 
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 R
      
     (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
      
                |  X0  X1  X2
     -----------+------------
      (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 BettiCharacters.m2:2846:0.