highestWeightsDecomposition(I,L)
highestWeightsDecomposition(I,deg)
highestWeightsDecomposition(I,lo,hi)
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 I is an ideal which is stable under the action of $G$.
Use this function to obtain the decomposition of a graded component of I. The input is the ideal I and the (multi)degree of the graded component.
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 $\mathbb{C}^3 \otimes \mathbb{C}^4$, with the natural action of $G = SL_3 (\mathbb{C}) \times SL_4 (\mathbb{C})$. The ideal is generated by the $2\times 2$ minors of a generic $3\times 4$ matrix.
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This shows that the component of I of degree 2 is the representation $S_{1,1} \mathbb{C}^3 \otimes S_{1,1} \mathbb{C}^4$. 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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