Example 1 -- The coordinate ring of the Grassmannian

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.

Now we assign weights to the variables of $R$. First we input the weights of $e_0,\ldots,e_5$ in a list L.

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.

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.

All monomials in $R$ are weight vectors. To recover the weight of a monomial, use the command getWeights with the monomial as the argument.

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 $R$.

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.

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.

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.