Let $E={\mathbb C}^6$, the standard representation of $SL_6 ({\mathbb C})$, with coordinate basis $\{e_0,\ldots,e_5\}$. The Grassmannian $V = Gr (2,E^*)$ is the projective variety which parametrizes two dimensional subspaces of $E^*$; it is embedded in $\PP (\wedge^2 E^*)$ using the Pl\"ucker equations (see Shafarevich - Basic Algebraic Geometry 1, Ch. I, Sec. 4.1). Consider $\wedge^2 E^*$ as a complex affine space. Let $C$ be the affine cone over $V$, i.e., the subvariety of $\wedge^2 E^*$ which is the union of all the one dimensional subspaces of $\wedge^2 E^*$ belonging to $V$. The space $\wedge^2 E^*$ has a natural action of $SL_6 ({\mathbb C})$ which fixes $C$.
Our polynomial ring $R$ is the ring of polynomial functions over $\wedge^2 E^*$, i.e., the symmetric algebra $Sym(\wedge^2 E)$. The elements $p_{i,j} = e_i \wedge e_j$, for $0\leq i < j \leq 5$, form a basis of weight vectors of $\wedge^2 E$ and will be the variables in $R$. The defining ideal of $C$ is generated by the Pl\"ucker equations; this ideal, which we call $I$, can be conveniently obtained in M2 using the command Grassmannian. We resolve the quotient $R/I$ as an $R$-module and call RI the minimal free resolution.
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Now we assign weights to the variables of $R$. First we input the weights of $e_0,\ldots,e_5$ in a list L.
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The weight of $p_{i,j} = e_i \wedge e_j$ is the sum of the weights of $e_i$ and $e_j$. The subscripts of the variables $p_{i,j}$ are the elements of subsets({0,1,2,3,4,5},2), the 2-subsets of the set $\{0,1,2,3,4,5\}$. Hence taking sums of pairs of weights in L over this indexing set will give us a complete list of weights for the variables $p_{i,j}$, as listed by M2.
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We declare D to be the Dynkin type $A_5$, which is the type of the group $SL_6 ({\mathbb C})$. Then we attach the weights in W to the variables in $R$ using the command setWeights; the arguments are in order the ring, the type and the weights of the variables.
The output will be the highest weight decomposition of the ${\mathbb C}$-linear subspace of $R$ generated by its variables; it is given in the form of a Tally, with keys describing the highest weights of the irreducible representation appearing in the decomposition and values equal to the multiplicities of those representations.
In this case, we get simply {0, 1, 0, 0, 0} => 1 which means that the decomposition contains only one copy of the irreducible representation with highest weight {0, 1, 0, 0, 0}, i.e., $\wedge^2 E$ as expected.
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All monomials in $R$ are weight vectors. To recover the weight of a monomial, use the command getWeights with the monomial as the argument.
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We can now issue the command highestWeightsDecomposition to obtain the decomposition of the representations corresponding to the free modules in the resolution; the only argument is the resolution RI. Suppose the free modules in RI are $F_0, \ldots, F_6$. The outermost HashTable in the output has keys equal to the subscripts of the free modules in RI. The value corresponding to a key $i$ is itself a HashTable with keys equal to the degrees of the generators of $F_i$. Finally the value corresponding to a certain degree $d$ is a Tally containing the highest weight decomposition of the representation $(F_i/m F_i)_d$ where $m$ is the maximal ideal generated by the variables in $R$.
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By analyzing the above output, we obtain the following description for RI: $$R \leftarrow \wedge^4 E \otimes R(-2) \leftarrow S_{2,1,1,1,1} E \otimes R(-3) \leftarrow S_{2} E \otimes R(-4) \oplus S_{2,2,2,2,2} E \otimes R(-5) \leftarrow S_{2,1,1,1,1} E \otimes R(-6) \leftarrow \wedge^2 E \otimes R(-7) \leftarrow R(-9) \leftarrow 0$$ Here $S_\lambda$ denotes the Schur functor associated to the partition $\lambda$ (for the construction of Schur functors see Fulton, Harris - Representation Theory, Ch. 6 or Fulton - Young Tableaux, Ch. 8). Recall that the Schur module $S_{\lambda} {\mathbb C}^{n+1}$ is the irreducible representation of $SL_{n+1} {\mathbb C}$ with highest weight $(\lambda_1 -\lambda_2,\ldots,\lambda_{n-1} -\lambda_n,\lambda_n)$ written in the basis of fundamental weights (as all lists of weights used by this package).
Next we turn to the coordinate ring of $C$, i.e., the quotient ring $Q=R/I$. We decompose its graded components in the range of degrees from 0 to 4, again with the command highestWeightsDecomposition. This time the arguments are the ring followed by the lowest and highest degrees in the range to be decomposed.
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We deduce that, for $d\in\{0,\ldots,4\}$, $(R/I)_d = S_{d,d} E$. We can also decompose the graded components of the ring $R$ in a range of degrees or in a single degree.
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For example, $R_2 = \wedge^4 E \oplus S_{2,2} E$. Since the representation $\wedge^4 E$ appears in $R_2$ but not in $(R/I)_2$, we deduce that it must be in $I_2$, the graded component of $I$ of degree 2. This can be verified directly by decomposing $I_2$ as follows.
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