splitCharacter -- Decomposes a symmetric polynomial as a sum of Schur functions

i1 : c=character({{1,1,1},{2}},4)

      3 3    4         3 2       2 3        4       3   2     2 2 2       3 2
o1 = x x  + x x x  + 2x x x  + 2x x x  + x x x  + 2x x x  + 2x x x  + 2x x x 
      0 1    0 1 2     0 1 2     0 1 2    0 1 2     0 1 2     0 1 2     0 1 2
     ------------------------------------------------------------------------
        3 3     2   3       2 3    3 3        4    4         3 2       2 3  
     + x x  + 2x x x  + 2x x x  + x x  + x x x  + x x x  + 2x x x  + 2x x x 
        0 2     0 1 2     0 1 2    1 2    0 1 2    0 1 3     0 1 3     0 1 3
     ------------------------------------------------------------------------
          4      4         3           2 2           3        4         3 2  
     + x x x  + x x x  + 4x x x x  + 5x x x x  + 4x x x x  + x x x  + 2x x x 
        0 1 3    0 2 3     0 1 2 3     0 1 2 3     0 1 2 3    1 2 3     0 2 3
     ------------------------------------------------------------------------
         2   2         2 2       3 2       2 3           3       2 3    
     + 5x x x x  + 5x x x x  + 2x x x  + 2x x x  + 4x x x x  + 2x x x  +
         0 1 2 3     0 1 2 3     1 2 3     0 2 3     0 1 2 3     1 2 3  
     ------------------------------------------------------------------------
        4        4       3   2     2 2 2       3 2     3   2     2     2  
     x x x  + x x x  + 2x x x  + 2x x x  + 2x x x  + 2x x x  + 5x x x x  +
      0 2 3    1 2 3     0 1 3     0 1 3     0 1 3     0 2 3     0 1 2 3  
     ------------------------------------------------------------------------
         2   2     3   2     2 2 2         2 2     2 2 2       3 2       3 2
     5x x x x  + 2x x x  + 2x x x  + 5x x x x  + 2x x x  + 2x x x  + 2x x x 
       0 1 2 3     1 2 3     0 2 3     0 1 2 3     1 2 3     0 2 3     1 2 3
     ------------------------------------------------------------------------
        3 3     2   3       2 3    3 3     2   3           3     2   3  
     + x x  + 2x x x  + 2x x x  + x x  + 2x x x  + 4x x x x  + 2x x x  +
        0 3     0 1 3     0 1 3    1 3     0 2 3     0 1 2 3     1 2 3  
     ------------------------------------------------------------------------
         2 3       2 3    3 3        4        4        4
     2x x x  + 2x x x  + x x  + x x x  + x x x  + x x x
       0 2 3     1 2 3    2 3    0 1 3    0 2 3    1 2 3

o1 : QQ[x ..x ]
         0   3

i2 : splitCharacter(c)

o2 = s      + s
      4,1,1    3,3

o2 : schurRing (QQ, s, 4)