splitCharacter -- decompose a symmetric polynomial into Schur functions

Usage:

splitCharacter(p)
Inputs:
- p, a ring element, a symmetric polynomial in the eigenvalues of the diagonal torus
Outputs:
- a ring element, the expansion of p as a $\mathbb{Z}_{\ge 0}$-linear combination of Schur functions $s_\lambda$

Description

Together with character this gives the isotypic decomposition of a polynomial $GL_d$-representation $W$ whose character is p: $$\chi_W = \sum_\lambda m_\lambda \cdot s_\lambda \;\Longleftrightarrow\; W \cong \bigoplus_\lambda S_\lambda(V)^{\oplus m_\lambda}.$$ The coefficients $m_\lambda$ are the multiplicities of the irreducible polynomial $GL_d$-representations in $W$.

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

i2 : splitCharacter c

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

o2 : schurRing (QQ, s, 4)

Verifies the Hermite-reciprocity / plethysm identity $\wedge^3(S^2 V_4) = S_{(4,1,1)}(V_4) \oplus S_{(3,3)}(V_4)$.

Ways to use splitCharacter:

splitCharacter(RingElement)

For the programmer

The object splitCharacter is a method function.

The source of this document is in /build/reproducible-path/macaulay2-1.26.05+ds/M2/Macaulay2/packages/SchurFunctors.m2:2690:0.

splitCharacter -- decompose a symmetric polynomial into Schur functions

Description

See also

Ways to use splitCharacter:

For the programmer