highestWeightsDecomposition(M,L,W)
highestWeightsDecomposition(M,deg,W)
highestWeightsDecomposition(M,lo,hi,W)
Let $G$ be a semisimple algebraic group which acts on a polynomial ring $R$ compatibly with the grading. Let $T\subseteq G$ be a maximal torus and assume the variables in $R$ are weight vectors for the action of $T$.
Suppose M is a finitely generated module over $R$ which is stable under the action of $G$. We assume that M was input in Macaulay2 using a specific presentation $\phi : E\rightarrow F$, where $E$ and $F$ are graded free $R$-modules of finite rank. Moreover assume the coordinate basis of $F$ is a homogeneous basis of weight vectors for the action of $T$ and the weight of the $j$-th basis vector is $w_j$.
Use this function to obtain the decomposition of a graded component of M. The input consists of three parameter: the module M, the (multi)degree of the graded component, and the list of weights $\{w_j\}$.
The output is a tally whose keys are the highest weights of certain irreducible representations and whose values are the multiplicities of those representations.
In the following example, the polynomial ring R is the symmetric algebra over $\mathbb{C}^2 \otimes \mathbb{C}^4$, with the natural action of $G=SL_2 (\mathbb{C}) \times SL_4 (\mathbb{C})$. The map $\phi$ is the unique (up to scalars) equivariant map $\mathbb{C}^4 \otimes R(-1) \rightarrow (\mathbb{C}^2)^* \otimes R$. If $\{f_1,f_2\}$ and $\{e_1,e_2,e_3,e_4\}$ are the coordinate bases of $\mathbb{C}^2$ and $\mathbb{C}^4$ respectively, then the matrix of $\phi$ with respect to the bases $\{f_1^*,f_2^*\}$ and $\{e_1,e_2,e_3,e_4\}$ is a generic matrix of indeterminates.
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This shows that the component of M of degree 2 is the representation $S_3 \mathbb{C}^3 \otimes S_2 \mathbb{C}^4$. Here $S_\lambda$ denotes the Schur functor corresponding to the partition $\lambda$.
When the polynomial ring is $\mathbb{Z}$-graded the degree can be given as an integer instead of a list. Moreover, in the $\mathbb{Z}$-graded case, one can decompose a range of degrees all at once as illustrated below.
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