characterRep -- extract the character of a polynomial GL-representation from its matrix

• Usage:

  characterRep F

• Inputs:
  • F, a matrix, a square matrix over a polynomial ring $R[w_1, \ldots, w_{d^2}]$ representing a polynomial $GL_d$-action on some module; typically the output of schur applied to a generic matrix
• Outputs:
  • a ring element, the trace of the diagonal specialization of F in a fresh polynomial ring $\mathbb{Q}[x_0, \ldots, x_{d-1}]$

i1 : R = QQ[w_1..w_9];

i2 : G = genericMatrix(R, 3, 3);

             3      3
o2 : Matrix R  <-- R

i3 : F = schur({2,1}, G);

             8      8
o3 : Matrix R  <-- R

i4 : characterRep F

      2        2    2                2        2      2
o4 = x x  + x x  + x x  + 2x x x  + x x  + x x  + x x
      0 1    0 1    0 2     0 1 2    1 2    0 2    1 2

o4 : QQ[x ..x ]
         0   2

i5 : splitCharacter characterRep F

o5 = s
      2,1

o5 : schurRing (QQ, s, 3)