character -- the character of a composition of Schur functors on GL(V)

Usage:

character(L, d)
Inputs:
- L, a list, a list of partitions $[\lambda^{(1)}, \lambda^{(2)}, \ldots, \lambda^{(m)}]$
- d, an integer, the rank of the underlying module
Outputs:
- a ring element, the character, as a symmetric polynomial in $x_0, \ldots, x_{d-1}$, of the composed Schur functor $S_{\lambda^{(1)}}(S_{\lambda^{(2)}}(\cdots S_{\lambda^{(m)}}(V)))$ on the defining representation $V = \mathbb{Q}^d$ of $GL_d$

Description

The character of a polynomial $GL_d$-representation $W$ is the trace of the diagonal matrix $\mathrm{diag}(x_0, \ldots, x_{d-1})$ acting on $W$. For an irreducible $S_\lambda(V)$ this is the Schur polynomial $s_\lambda(x_0, \ldots, x_{d-1})$; for an arbitrary polynomial representation it is a non-negative integer combination of Schur polynomials, making character the primary input to splitCharacter and decomposeRep.

i1 : character({{2,1}}, 3)

      2        2    2                2        2      2
o1 = 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

o1 : QQ[x ..x ]
         0   2

Nested compositions are supported; for example the GL_4 character of $\wedge^3(S^2 V) = S_{(1,1,1)}(S_{(2)}(V))$:

i2 : character({{1,1,1}, {2}}, 4)

      3 3    4         3 2       2 3        4       3   2     2 2 2       3 2
o2 = 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

o2 : QQ[x ..x ]
         0   3

To split such a character into irreducibles, apply splitCharacter:

i3 : splitCharacter character({{1,1,1}, {2}}, 4)

o3 = s      + s
      4,1,1    3,3

o3 : schurRing (QQ, s, 4)

Ways to use character:

character(List,ZZ)

For the programmer

The object character is a method function.

The source of this document is in /build/reproducible-path/macaulay2-1.26.05+ds/M2/Macaulay2/packages/SchurFunctors.m2:2619:0.

character -- the character of a composition of Schur functors on GL(V)

Description

See also

Ways to use character:

For the programmer