Schubert calculus

Schubert calculus can be constructed using the Chow ring Denote the Grassmannian of k-planes in a fixed n-dimensional vector space V as \mathbf{Gr}(k,V), and its Chow ring as A^*(\mathbf{Gr}(k,V)). (Note that the Grassmannian is sometimes denoted \mathbf{Gr}(k,n) if the vector space isn't explicitly given or as \mathbb{G}(k-1,n-1) if the ambient space V and its k-dimensional subspaces are replaced by their projectivizations.) Choosing an (arbitrary) complete flag ::\mathcal{V} = (V_1 \subset \cdots \subset V_{n-1} \subset V_n = V), \quad \dim{V}_i = i, \quad i=1 , \dots , n, to each weakly decreasing k-tuple of integers \mathbf{a} = (a_1,\ldots, a_k), where ::n-k \geq a_1 \geq a_2 \geq \cdots \geq a_k \geq 0, i.e., to each partition of weight :: |\mathbf{a}| = \sum_{i=1}^k a_i, whose Young diagram fits into the k \times (n-k) rectangular one for the partition (n-k)^k, we associate a Schubert variety Properties Inclusion There is a partial ordering on all k-tuples where \mathbf{a} \geq \mathbf{b} if a_i \geq b_i for every i. This gives the inclusion of Schubert varieties :: \Sigma_{\mathbf{a}} \subset \Sigma_{\mathbf{b}} \iff \mathbf{a} \geq \mathbf{b}, showing an increase of the indices corresponds to an even greater specialization of subvarieties. Dimension formula A Schubert variety \Sigma_{\mathbf{a}} has codimension equal to the weight ::|\mathbf{a}|=\sum a_i of the partition \mathbf{a}. Alternatively, in the notational convention S_\lambda indicated above, its dimension in \mathbf{Gr}(k,n) is the weight :: |\lambda|= \sum_{i=1}^k\lambda_i = k(n-k)-|\mathbf{a}|. of the complementary partition \lambda \subset (n-k)^k in the k \times (n-k) dimensional rectangular Young diagram. This is stable under inclusions of Grassmannians. That is, the inclusion :: i_{(k, n)}:\mathbf{Gr}(k, \mathbf{C}^n) \hookrightarrow \mathbf{Gr}(k, \mathbf{C}^{n+1}), \quad \mathbf{C}^n =\text{span}\{e_1, \dots, e_n\} defined, for w\in\mathbf{Gr}(k, \mathbf{C}^n) , by ::i_{(k, n)}: w \subset \mathbf{C}^n \mapsto w \subset \mathbf{C}^n\oplus \mathbf {C} e_{n+1} =\mathbf{C}^{n+1} has the property ::i^*_{(k, n)}(\sigma_{\mathbf{a}}) = \sigma_{\mathbf{a}}, and the inclusion :: \tilde{i}_{(k,n)}: \mathbf{Gr}(k,n) \hookrightarrow \mathbf{Gr}(k+1,n+1) defined by adding the extra basis element e_{n+1} to each k-plane, giving a (k+1)-plane, ::\tilde{i}_{(k, n)}: w \mapsto w \oplus\mathbf {C} e_{n+1} \subset \mathbf{C}^n \oplus \mathbf {C} e_{n+1} =\mathbf{C}^{n+1} does as well ::\tilde{i}_{(k,n)}^*(\sigma_{\mathbf{a}}) = \sigma_{\mathbf{a}}. Thus, if X_{\mathbf{a}} \subset \mathbf{Gr}_k(n) and \Sigma_{\mathbf{a}} \subset \mathbf{Gr}_k(n) are a cell and a subvariety in the Grassmannian \mathbf{Gr}_k(n), they may also be viewed as a cell X_{\mathbf{a}} \subset \mathbf{Gr}_\tilde{k}(\tilde{n}) and a subvariety \Sigma_{\mathbf{a}} \subset \mathbf{Gr}_\tilde{k}(\tilde{n}) within the Grassmannian \mathbf{Gr}_\tilde{k}(\tilde{n}) for any pair (\tilde{k}, \tilde{n}) with \tilde{k} \geq k and \tilde{n}-\tilde{k} \geq n-k. Intersection product The intersection product was first established using the Pieri and Giambelli formulas. Pieri formula In the special case \mathbf{b} = (b,0,\ldots, 0), there is an explicit formula of the product of \sigma_b with an arbitrary Schubert class \sigma_{a_1,\ldots, a_k} given by ::\sigma_b\cdot\sigma_{a_1,\ldots, a_k} = \sum_{ \begin{matrix}|c| = |a| + b \\ a_i \leq c_i \leq a_{i-1} \end{matrix} } \sigma_{\mathbf{c}}, where |\mathbf{a}| = a_1 + \cdots + a_k, |\mathbf{c}| = c_1 + \cdots + c_k are the weights of the partitions. This is called the Pieri formula, and can be used to determine the intersection product of any two Schubert classes when combined with the Giambelli formula. For example, ::\sigma_1 \cdot \sigma_{4,2,1} = \sigma_{5,2,1} + \sigma_{4,3,1} + \sigma_{4,2,1,1}. and ::\sigma_2 \cdot \sigma_{4,3} = \sigma_{4,3,2} + \sigma_{4,4,1} + \sigma_{5,3,1} + \sigma_{5,4} + \sigma_{6,3} Giambelli formula Schubert classes \sigma_{\mathbf{a}} for partitions of any length \ell(\mathbf{a})\leq k can be expressed as the determinant of a (k \times k) matrix having the special classes as entries. :: \sigma_{(a_1,\ldots, a_k)} = \begin{vmatrix} \sigma_{a_1} & \sigma_{a_1 + 1} & \sigma_{a_1 + 2} & \cdots & \sigma_{a_1 + k - 1} \\ \sigma_{a_2 - 1} & \sigma_{a_2} & \sigma_{a_2 + 1} & \cdots & \sigma_{a_2 + k - 2} \\ \sigma_{a_3 - 2} & \sigma_{a_3 - 1} & \sigma_{a_3} & \cdots & \sigma_{a_3 + k - 3} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ \sigma_{a_k - k + 1} & \sigma_{a_k - k + 2} & \sigma_{a_k - k + 3} & \cdots & \sigma_{a_k} \end{vmatrix} This is known as the Giambelli formula. It has the same form as the first Jacobi-Trudi identity, expressing arbitrary Schur functions s_{\mathbf{a}} as determinants in terms of the complete symmetric functions \{h_j := s_{(j)}\}. For example, ::\sigma_{2,2} = \begin{vmatrix} \sigma_2 & \sigma_3 \\ \sigma_1 & \sigma_2 \end{vmatrix} = \sigma_2^2 - \sigma_1\cdot\sigma_3 and ::\sigma_{2,1,1} = \begin{vmatrix} \sigma_2 & \sigma_3 & \sigma_4 \\ \sigma_0 & \sigma_1 & \sigma_2 \\ 0 & \sigma_0 & \sigma_1 \end{vmatrix}. General case The intersection product between any pair of Schubert classes \sigma_{\mathbf{a}}, \sigma_{\mathbf{b}} is given by :: \sigma_{\mathbf{a}} \sigma_{\mathbf{b}} =\sum_{\mathbf{c}}c^\mathbf{c}_{\mathbf{a} \mathbf{b}}\sigma_{\mathbf{c}}, where \{c^\mathbf{c}_{\mathbf{a} \mathbf{b}}\} are the Littlewood-Richardson coefficients. The Pieri formula is a special case of this, when \mathbf{b}=(b,0, \dots, 0) has length \ell(\mathbf{b})=1. == Relation with Chern classes ==

There is an easy description of the cohomology ring, or the Chow ring, of the Grassmannian \mathbf{Gr}(k,V) using the Chern classes of two natural vector bundles over \mathbf{Gr}(k,V). We have the exact sequence of vector bundles over \mathbf{Gr}(k,V) ::0 \to T \to \underline{V} \to Q \to 0 where T is the tautological bundle whose fiber, over any element w \in \mathbf{Gr}(k, V) is the subspace w \subset V itself, \, \underline{V}:= \mathbf{Gr}(k,V) \times V is the trivial vector bundle of rank n, with V as fiber and Q is the quotient vector bundle of rank n-k, with V/w as fiber. The Chern classes of the bundles T and Q are ::c_i(T) = (-1)^i\sigma_{(1)^i}, where (1)^i is the partition whose Young diagram consists of a single column of length i and ::c_i(Q) = \sigma_i. The tautological sequence then gives the presentation of the Chow ring as ::A^*(\mathbf{Gr}(k,V)) = \frac{\mathbb{Z}[c_1(T), \ldots, c_k(T), c_1(Q),\ldots, c_{n-k}(Q)] }{(c(T)c(Q) - 1)}. == Gr(2,4) ==

One of the classical examples analyzed is the Grassmannian \mathbf{Gr}(2,4) since it parameterizes lines in \mathbb{P}^3. Using the Chow ring A^*(\mathbf{Gr}(2,4)), Schubert calculus can be used to compute the number of lines on a cubic surface. it is given by :: \begin{align} A^0(\mathbf{Gr}(2,4)) &= \mathbb{Z}\cdot 1 \\ A^2(\mathbf{Gr}(2,4)) &= \mathbb{Z}\cdot \sigma_1 \\ A^4(\mathbf{Gr}(2,4)) &= \mathbb{Z}\cdot \sigma_2 \oplus \mathbb{Z} \cdot \sigma_{1,1}\\ A^6(\mathbf{Gr}(2,4)) &= \mathbb{Z}\cdot\sigma_{2,1} \\ A^8(\mathbf{Gr}(2,4)) &= \mathbb{Z}\cdot\sigma_{2,2} \\ \end{align} Lines on a cubic surface Recall that a line in \mathbb{P}^3 gives a dimension 2 subspace of \mathbb{A}^4, hence an element of \mathbb{G}(1,3) \cong \mathbf{Gr}(2,4). Also, the equation of a line can be given as a section of \Gamma(\mathbb{G}(1,3), T^*). Since a cubic surface X is given as a generic homogeneous cubic polynomial, this is given as a generic section s \in \Gamma(\mathbb{G}(1,3),\text{Sym}^3(T^*)). A line L \subset \mathbb{P}^3 is a subvariety of X if and only if the section vanishes on [L] \in \mathbb{G}(1,3). Therefore, the Euler class of \text{Sym}^3(T^*) can be integrated over \mathbb{G}(1,3) to get the number of points where the generic section vanishes on \mathbb{G}(1,3). In order to get the Euler class, the total Chern class of T^* must be computed, which is given as ::c(T^*) = 1 + \sigma_1 + \sigma_{1,1} The splitting formula then reads as the formal equation ::\begin{align} c(T^*) &= (1 + \alpha)(1 + \beta) \\ &= 1 + \alpha + \beta + \alpha\cdot\beta \end{align}, where c(\mathcal{L}) = 1+\alpha and c(\mathcal{M}) = 1 + \beta for formal line bundles \mathcal{L},\mathcal{M}. The splitting equation gives the relations ::\sigma_1 = \alpha + \beta and \sigma_{1,1} = \alpha\cdot\beta. Since \text{Sym}^3(T^*) can be viewed as the direct sum of formal line bundles ::\text{Sym}^{3}(T^{*}) = \mathcal{L}^{\otimes 3} \oplus (\mathcal{L}^{\otimes 2} \otimes \mathcal{M}) \oplus(\mathcal{L}\otimes\mathcal{M}^{\otimes 2})\oplus \mathcal{M}^{\otimes 3} whose total Chern class is ::c(\text{Sym}^3(T^*)) = (1 + 3\alpha)(1 + 2\alpha + \beta)(1 + \alpha + 2\beta)(1 + 3\beta), it follows that ::\begin{align} c_4(\text{Sym}^3(T^*)) &= 3\alpha (2\alpha + \beta) (\alpha + 2\beta) 3\beta \\ &=9\alpha\beta(2(\alpha + \beta)^2 + \alpha\beta) \\ &= 9\sigma_{1,1}(2\sigma_1^2 + \sigma_{1,1}) \\ &= 27\sigma_{2,2}\, , \end{align} using the fact that ::\sigma_{1,1}\cdot \sigma_1^2 = \sigma_{2,1}\sigma_1 = \sigma_{2,2} and \sigma_{1,1}\cdot \sigma_{1,1} = \sigma_{2,2}. Since \sigma_{2,2} is the top class, the integral is then ::\int_{\mathbb{G}(1,3)}27\sigma_{2,2} = 27. Therefore, there are 27 lines on a cubic surface. == See also ==