Posted by: Steven Sam | December 7, 2009

Set-valued tableaux and Grothendieck polynomials

Last time, I discussed a bit about the relationship between the Chow groups of a nonsingular variety X and its K-theory. In this post, I want to specialize to the case when X is a Grassmannian and explain some of the combinatorics behind this relationship.

Set-valued tableaux

For a partition \lambda, we let D(\lambda) denote its Young diagram, i.e., we draw \lambda_i boxes in the ith row (counting from top to bottom) all left-justified. Also |\lambda| denotes the sum of its parts. Given two partitions \lambda, \mu, with \lambda \supseteq \mu (i.e., \lambda_i \ge \mu_i for all i), we set D(\lambda/\mu) = D(\lambda) \setminus D(\mu) denote its Young diagram.

Given two finite subsets A and B of positive integers, denote A<B if max(A) < min(B) and A \le B if \max(A) \le \min(B). This is not meant to define a partial order.

A set-valued tableau of D(\lambda) is an assignment T of a finite subset of positive integers to each cell in such a way that T(i,j) \le T(i,j+1) and T(i,j) < T(i-1,j) using the notation for subsets defined above. Given a set-valued tableau T, we associate to it a monomial x^T = x_1^{T_1} x_2^{T_2} \cdots where T_i is the number of times that the integer i appears in a cell of T. Also, let |T| denote the total degree of this monomial. We define the (single stable) Grothendieck polynomial to be

\displaystyle G_{\lambda/\mu}(x) = \sum_T (-1)^{|T|-|\lambda|-|\mu|} x^T

where the sum is over all set-valued tableaux of D(\lambda/\mu). Then G_{\lambda/\mu}(x) is a symmetric function in the x_i. This is not obvious, but one can prove it to be true purely combinatorially in the same way one can prove that Schur functions are symmetric purely combinatorially (it is enough to show that it is invariant under switching x_i and x_{i+1}, and this can be shown by directly switching the number of i’s and (i+1)’s within any given set-valued tableau).

In the case that each subset has size 1, the definition of a set-valued tableau specializes to the notion of a semistandard Young tableau, so the lowest degree term of G_{\lambda/\mu}(x) is the skew Schur function s_{\lambda/\mu}(x) (by definition, if you like). Let \Gamma be the subring of the symmetric functions generated by the G_{\lambda/\mu}(x). Since the Schur functions s_\lambda(x) are linearly independent as we vary over all partitions, the same is true for the G_\lambda(x) (in fact, they form a basis).

K-theory of the Grassmannian

Let X be Gr(k,n), the Grassmannian of k-planes in an n-dimensional vector space V. Let \mathcal{R} \subset V \times X be the tautological subbundle of rank k on X, i.e., the fiber of \mathcal{R} over a point (which is a subspace of V) of X is identified with the subspace itself, and let \mathcal{R}^\vee denote its dual.

Given a vector bundle E which is a direct sum of line bundles E = L_1 \oplus \cdots \oplus L_e, we set

G_{\lambda/\mu}(E) = G_{\lambda/\mu}(1 - L_1^\vee, \dots, 1-L_e^\vee, 0, 0, \dots)

as an element in the K-theory K(X). Since G_{\lambda/\mu}(x) is symmetric, this is well-defined. Furthermore, this implies that G_{\lambda/\mu}(E) is a polynomial in the elementary symmetric functions in the variables 1-L_i^\vee. But each of these elementary symmetric functions can be expressed solely in terms of an appropriate exterior power of E. Hence, we can make this definition for arbitrary E that do not split up as a sum of line bundles.

We define a map \Gamma \to {\rm K}(X) by G_\lambda(x) \mapsto G_\lambda(\mathcal{R}^\vee). This map is surjective, and its kernel is the ideal generated by G_\mu such that the partition \mu does not fit inside the k \times (n-k) rectangular partition.

Cohomology of the Grassmannian

We have a similar situation when we study the cohomology ring (equivalently in this case, the Chow ring) of X. Fix a basis \{e_1, \dots, e_n\} for V. Let B be the subgroup of upper triangular matrices with respect to this basis. For a partition \lambda fitting inside the k \times (n-k) rectangle, define a point e_\lambda in the Grassmannian to be the subspace \langle e_{n-k+1-\lambda_1}, e_{n-k+2-\lambda_2} \dots, e_{n-\lambda_k} \rangle, and define the Schubert variety to be the closure of the B orbit of e_\lambda. It has codimension |\lambda| inside of the Grassmannian. Let \sigma_\lambda \in {\rm H}^{|\lambda|}(X) denote the Poincaré dual of the Schubert variety X_\lambda. Let \Lambda denote the ring of symmetric functions. Then the map \Lambda \to {\rm H}^*(X) defined by s_\lambda(x) \mapsto \sigma_\lambda is surjective, and the kernel is the ideal generated by s_\mu(x) for \mu which do not fit inside the k \times (n-k) rectangular partition.

Relationship between K-theory and cohomology

Combinatorially, we can see that the associated graded of the K-theory coincides with the cohomology of the Grassmannian. To be precise, we can filter the ring \Gamma by the ideals \Gamma_m = \sum_{|\lambda| \ge m} \langle G_\lambda \rangle. Then the Grothendieck polynomials get turned into their lowest degree terms, which we saw above are just the corresponding Schur functions.

Geometrically, we have the following. If \sigma_m = [X_m] is the cohomology class of the Schubert variety corresponding to the partition with one part equal to m, then \sigma_m is the mth Chern class of the quotient bundle \mathcal{Q} = (V \times X) / \mathcal{R}. The mth Chern class of \mathcal{Q} is the mth elementary symmetric function in the Chern roots of \mathcal{Q}; let h_m(\mathcal{Q}) denote the mth homogeneous symmetric function in the Chern roots. In general one has the identity

\displaystyle \sum_{i+j=m} (-1)^j e_i h_j = 0 (m > 0),

and from the short exact sequence 0 \to \mathcal{R} \to V \times X \to \mathcal{Q} \to 0, we get that

\displaystyle \sum_{i+j=m} e_i(\mathcal{R}) e_j(\mathcal{Q}) = 0 (m > 0),

since the Chern classes of a trivial vector bundle are 0. Hence we conclude that e_j(\mathcal{Q}) = (-1)^j h_j(\mathcal{R}) for all j.

The map from cohomology to the associated graded of K-theory sends the Chern class of a line bundle L to the class 1-[L^\vee]. This follows from the definition given last time, the short exact sequence 0 \to L^\vee \to \mathcal{O}_X \to \mathcal{O}_D \to 0 when D is an effective divisor and L is the line bundle associated with D, and the fact that c_1(L) = [D] in cohomology, and then extending linearly to all divisors.

So we see that [\sigma_m] = e_m(\mathcal{Q}) = (-1)^m h_m(\mathcal{R}) = h_m(\mathcal{R}^\vee) gets sent to the image of G_m(\mathcal{R}^\vee) in the associated graded of K(X). Now we can deduce the general case by quoting the fact that the cohomology ring of the Grassmannian is generated by the classes \sigma_m.

So everything matches up! There is more I could say about the combinatorics of Grothendieck polynomials, but I’ll stop here.



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