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This article is cited in 1 scientific paper (total in 1 paper) Schubert calculus and intersection theory of flag manifolds H. Duanabc, X. Zhaod a Yau Mathematical Sciences Center, Tsinghua University, Beijing, China b Academy of Mathematics and Systems Sciences, Chinese Academy of Sciences, Beijing, China c School of Mathematical Sciences, Dalian University of Technology, Dalian, China d Department of Mathematics, Capital Normal University, Beijing, China
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     1. IntroductionHilbert’s 15th problem [43] is an inspiring and far-reaching one. It promoted the growth of the enumerative geometry of the 19th century into algebraic geometry founded by Van der Waerden and Weil, and integrated Schubert calculus deeply into many branches of mathematics. However, despite a great many achievements in the 20th century (see, for example, [38], [49], and [65]), the part of the problem on finding an effective rule for performing calculus has been stagnant for a long time, notably, the Schubert problem of characteristics (see [68], § 8), or the Weil problem (see [70], p. 331) of the intersection theory of flag manifolds $G/P$, where $G$ is a compact connected Lie group and $P$ is a parabolic subgroup [4]. In the series of works [19], [20], [28], [29], and [31], we addressed both the problem of characteristics and the Weil problem, with the implication that the 15th problem was solved satisfactorily (see [32], Remark 6.3). The purpose of this article is to give an overview of the background, content, and the solution of the Schubert problem of characteristics. In § 2 we glimpse the development from Apollonius’s work “Tangencies” to Lefschetz’s homology theory, which reflects the evolution of the basic ideas from enumerative geometry to intersection theory. Section 3 summarizes the pioneer contributions of Van der Waerden, Ehresmann, Weil, and Bernstein–Gel’fand–Gel’fand to Schubert calculus, which have led to a great clarification of Schubert characteristics. Our main results are introduced in § 4, where we present a formula expressing the Shubert characteristics of a flag manifold $G/P$ as polynomials in the Cartan numbers of the Lie group $G$ (Theorem 4.5), develop a systematic description of the intersection rings of flag manifolds (Theorem 4.8), and illustrate the effectiveness of our computer programs by Examples 4.6, 4.9, and 4.11. In particular, since our approach uses the Cartan matrices of Lie groups as the main input, the solutions can be implemented successfully via computer programs, so that the intersection theory of flag manifolds becomes easily accessible to a broad range of readers. 2. An introduction to intersection theoryIn the 2nd century BC Apollonius of Perga obtained the following enumerative result in his paper “Tangencies”. The original proof of Apollonius was lost, but a record of the theorem by Pappus, dated the 4th century, survived. During the Renaissance, different proofs were found by Viète, van Roomen, Gergonne, and Newton (see [10], p. 159). For a pictorial illustration of the theorem, see the cover-page story of the book 3264 and all that [34]. Descartes’s discovery of Euclidean coordinates made it possible for geometers (for example, Maclaurin, Euler, Bezout) to exploit polynomial systems to characterize geometric figures that satisfy a system of incidence conditions. Consequently, many enumerative problems admit the following algebraic formulation. Problem 2.1. Given a system of $n$ polynomials in $n$ variables with complex coefficients find the number of solutions to this system. Problem 2.1 is a fundamental one in algebra. In the case $n=1$ Gauss proved in the 1820s that the number of zeros of a polynomial in a single variable is the degree of this polynomial, which is well known as the fundamental theorem of algebra. Letting $g_{i}$ be the homogenization of the polynomial $f_{i}$ in (2.1), in the $n$- dimensional complex projective space $\mathbb{C}\mathbb{P}^n$ we obtain a hypersurface $N_{i}:=g_{i}^{-1}(0)$. In general, the zero locus of a homogeneous system on a complex projective space is called a projective variety. Naturally, the study of Problem 2.1 leads to the fundamental problem of intersection theory. Problem 2.2. Given $k$ subvarieties $N_{1},\dots,N_{k}$ in a smooth projective manifold $M$ that satisfy the dimension constraint $\dim N_{1}+\cdots +\dim N_{k}=(k-1)\dim M$, find the number $|N_1\cap\cdots\cap N_k|$ of their common points of intersection when the subvarieties $N_{i}$ are in general position (see Fig. 1). In the course of studying Problem 2.2 Lefschetz developed the homology theory for cellular complexes [51] (1926). In the perspective of this theory, let $\alpha_{i}\in H^{\dim M-\dim N_{i}}(M)$ be the Poincaré dual of the cycle class represented by the subvariety $N_{i}\subset M$. Problem 2.3. Given $k$ projective subvarieties $N_{1},\dots,N_{k}$ in a smooth projective manifold $M$ that satisfy the dimension constraint $\dim N_{1}+\cdots +\dim N_{k}=({k-1})\dim M$, compute the Kronecker pairing where $\cup$ means the cup product on the cohomology ring $H^*(M)$ and $[M]$ denotes the orientation class of $M$. Through Problems 2.1–2.3 we have briefly reviewed three seemingly different approaches to problems of enumerative geometry. Given that the effective computability is the primary task of enumerative geometry, a natural question is: which approach is the most calculable one? The development of intersection theory shall tell us the answer. 3. Schubert problem of characteristicsHermann Schubert (1848–1911) received his Ph.D. from the University of Halle, Germany, in 1870. His doctoral thesis The theory of characteristics [58] was about enumerative geometry. Prior to this he had shown that there are $16$ spheres tangent to $4$ general spheres in space, which is a direct extension of Apollonius’s theorem. In 1879 Schubert published the celebrated book Calculus of enumerative geometry [61], which presents the summit of intersection theory in the late 19th century (see [34], p. 2). While developing Chasles’s work on conics [13], he demonstrated amazing applications of intersection theory to enumerative geometry, such as
In order to gain insight into the central part of Schubert’s approach to those spectacular enumerative numbers, we resort to the table of characteristics (see Table 1) for the variety of complete conics in space from his book (see [61], p. 95), where the symbols $\mu$, $\nu$, and $\rho$ stand for the subvarieties of conics passing through a given point, intersecting a given line, and tangent to a given plane, respectively. Table 1.The characteristics of the space of complete conics in $\mathbb{C}\mathbb{P}^3$
The table consists of equalities equating the monomials $\mu^{m}\nu^{n}\rho ^{8-m-n}$ to integers, which were called characteristics by Schubert and Schubert’s symbolic equations by earlier researchers. Schubert emphasized that the problem of characteristics is a fundamental one in enumerative geometry [48], [58], [61], [60]. However, to state this problem in its natural simplicity and generality, one had to wait until the 1950s for the celebrated ‘basis theorem of Schubert calculus’. Let us recall relevant works on the subject. The study of characteristics began with the Italian school headed by Segre, Enriques, and Severi. Two representative papers of this school were “The principle of conservation of numbers” and “The foundation of enumerative geometry and the theory of characteristics” by Severi [62], [63]. Regarding these works, Van de Wareden [69] commented that they “erected an admirable structure, but its logical foundation was shaky. The notions were not well-defined, the proofs were insufficient”. In the pioneering work “Topological foundation of enumerative geometry” [68] (1930) Van der Waerden interpreted Schubert characteristics in the perspective of homology theory developed by Lefschetz [51] (see Problem 2.3). He made the following crucial observations, which enlightened the course of the later studies on the 15th problem:
In 1934 Ehresmann [33] went two important steps further: he discovered that
In what follows we denote by $W(P;G)$ the set of left cosets of the Weyl group $W(G)$ of $G$ by the Weyl group $W(P)$ of $P$, and let $l\colon W(P;G)\to \mathbb{Z}$ denote the associated length function [4]. With the in-depth research on the structure of Lie groups (see, for instance, [7]) the vague term ‘Schubert symbols’ in the early literature was gradually replaced by such rigorously defined geometric objects as ‘Schubert cells’ or ‘Schubert varieties’. In particular, extending Ehresmann’s work [33] to the Grassmannian $G_{n,k}$, the following result was announced by Chevalley [14] for complete flag manifolds $G/T$ (where $T\subset G$ is a maximal torus) and extended by Bernstein, I. Gel’fand, and S. Gel’fand to all flag manifolds $G/P$ (see [4], Proposition 5.1). Theorem 3.1. Every flag manifold $G/P$ has a canonical decomposition into Schubert cells $S_{w}$ parameterized by elements $w$ of $W(P;G)$, where the closure $X_{w}$ of each cell $S_{w}$ is a subvariety of $G/P$, called the Schubert variety in $G/P$ associated to $w\in W(P;G)$. Since only even-dimensional cells are involved in the partition (3.1), the set $\{[X_{w}],\, w\in W(P;G)\}$ of fundamental classes forms an additive basis of the homology $H_*(G/P)$. The co-cycle classes $s_{w}\in H^*(G/P)$ Kronecker dual to this basis (that is, $\langle s_w,[X_{u}]\rangle =\delta_{w,u}$, $w,u\in W(P;G)$) are called the Schubert classes associated to $w\in W(P;G)$. Theorem 3.1 implies the following result, which was expected by Van der Waerden (see [68], § 8), and is well known as the ‘basis theorem of Schubert calculus’. Theorem 3.2 (see [4], Proposition 5.2). The set $\{s_{w},\, w\in W(P;G)\}$ of Schubert classes forms a basis of the cohomology $H^*(G/P)$. An immediate consequence of the basis theorem is that any product $s_{u_{1}}\cdots s_{u_{k}}$ of Schubert classes can uniquely be expressed as a linear combination of the basis elements:
$$
\begin{equation}
s_{u_1}\cdots s_{u_k} =\sum_{ w\in W(P;G),\, l(w)=l(u_1)+\cdots +l(u_k) } c_{u_1,\dots,u_k}^w\cdot s_w,
\end{equation}
\tag{3.2}
$$
so that the Schubert problem of characteristics [33], [59], [60], [68] has the following concise expression. 1 Problem 3.3. Given any monomial $s_{u_{1}}\cdots s_{u_{k}}$ in the Schubert classes, determine the characteristics numbers $c_{u_{1},\dots,u_{k}}^{w}$ in the linear expansion (3.2). In the momentous treatise Foundations of Algebraic Geometry [70] Weil completed the definition of intersection multiplicities for the first time, and summarized the task of Schubert calculus in the context of modern intersection theory. 2 Weil commented his problem as ‘the modern form taken by the topic formerly known as enumerative geometry’ (see [70], p. 331). We show the following. Proof. A ring is an abelian group $R$ that is furnished with multiplication $R\times R\to R$. By the basis theorem (Theorem 3.2) the cohomology $H^*(G/P)$ has a canonical basis consisting of Schubert classes. Therefore, multiplication on $H^*(G/P)$ is uniquely determined by the products of basis elements, which are governed by the characteristics $c_{u_{1},\dots,u_{k}}^{w}$. $\Box$ Remark 3.6. In the case $k=2$ the characteristics $c_{w_{1},\dots,w_{k}}^{w}$ admit various interpretations. They are called Schubert structure constants of the flag manifold $G/P$ in topology and Littlewood–Richardson coefficients in representation theory [52]. In certain cases the parameter spaces of geometric figures considered by Schubert (see [61], Chap. IV) fail to be flag manifolds, but can be constructed by performing a finite number of steps of blow-ups on flag manifolds (see examples in Fulton [38], § 10.4, Eisenbud and Harris [34], Chap. 13, or in [23] for the constructions of the parameter spaces of complete conics and quadrics in $\mathbb{C}\mathbb{P}^3$). As a result, the relevant characteristics can be computed from those of flag manifolds via strict transformations (see, for instance, [23], Examples 5.11 and 5.12). 4. Intersection theory of flag manifoldsTo secure the foundation of a ‘calculus’ it suffices to decide the objects to be calculated and to determine accordingly the rules of calculation (see, for example, [2], Chap. 2, and [53]). As for Schubert calculus, we saw in § 3 that the objects to be calculated have been clarified to be the Schubert symbols, or Schubert varieties. In this section we determine the rules of calculus by a unified formula computing the characteristics, and apply this formula to complete the intersection theory of flag manifolds. 4.1. An observation and an expectationThe major difficulties that one encounters with the problem of characteristics are fairly transparent:
Summarizing, studies case by case can never reach a complete solution to the problem. On the other hand, according to É. Cartan’s beautiful work on compact Lie groups, associated to each simple Lie group $G$ there is a Cartan matrix $C$, which plays the role of the ‘cosmological constants’ to classify all flag manifolds $G/P$ (see the discussions in §§ 4.2–4.5). For example, for the five exceptional Lie groups these matrices are
$$
\begin{equation*}
G_2\colon\begin{pmatrix} \hphantom{-} 2 & -1 \\ -3 & \hphantom{-} 2 \end{pmatrix}, \quad F_4\colon\begin{pmatrix} \hphantom{-} 2 & -1 & \hphantom{-} 0 & \hphantom{-} 0 \\ -1 & \hphantom{-} 2 & -2 & \hphantom{-} 0 \\ \hphantom{-} 0 & -1 & \hphantom{-} 2 & -1 \\ \hphantom{-} 0 & \hphantom{-} 0 & -1 & \hphantom{-} 2 \end{pmatrix}, \quad E_6\colon\begin{pmatrix} \hphantom{-} 2 & \hphantom{-} 0 & -1 & \hphantom{-} 0 & \hphantom{-} 0 & \hphantom{-} 0 \\ \hphantom{-} 0 & \hphantom{-} 2 & \hphantom{-} 0 & -1 & \hphantom{-} 0 & \hphantom{-} 0 \\ -1 & \hphantom{-} 0 & \hphantom{-} 2 & -1 & \hphantom{-} 0 & \hphantom{-} 0 \\ \hphantom{-} 0 & -1 & -1 & \hphantom{-} 2 & -1 & \hphantom{-} 0 \\ \hphantom{-} 0 & \hphantom{-} 0 & \hphantom{-} 0 & -1 & \hphantom{-} 2 & -1 \\ \hphantom{-} 0 & \hphantom{-} 0 & \hphantom{-} 0 & \hphantom{-} 0 & -1 & \hphantom{-} 2 \end{pmatrix},
\end{equation*}
\notag
$$
$$
\begin{equation*}
E_7\colon\begin{pmatrix} \hphantom{-} 2 & \hphantom{-} 0 & -1 & \hphantom{-} 0 & \hphantom{-} 0 & \hphantom{-} 0 & \hphantom{-} 0 \\ \hphantom{-} 0 & \hphantom{-} 2 & \hphantom{-} 0 & -1 & \hphantom{-} 0 & \hphantom{-} 0 & \hphantom{-} 0 \\ -1 & \hphantom{-} 0 & \hphantom{-} 2 & -1 & \hphantom{-} 0 & \hphantom{-} 0 & \hphantom{-} 0 \\ \hphantom{-} 0 & -1 & -1 & \hphantom{-} 2 & -1 & \hphantom{-} 0 & \hphantom{-} 0 \\ \hphantom{-} 0 & \hphantom{-} 0 & \hphantom{-} 0 & -1 & \hphantom{-} 2 & -1 & \hphantom{-} 0 \\ \hphantom{-} 0 & \hphantom{-} 0 & \hphantom{-} 0 & \hphantom{-} 0 & -1 & \hphantom{-} 2 & -1 \\ \hphantom{-} 0 & \hphantom{-} 0 & \hphantom{-} 0 & \hphantom{-} 0 & \hphantom{-} 0 & -1 & \hphantom{-} 2 \end{pmatrix}, \quad E_{8}\colon\begin{pmatrix} \hphantom{-} 2 & \hphantom{-} 0 & -1 & \hphantom{-} 0 & \hphantom{-} 0 & \hphantom{-} 0 & \hphantom{-} 0 & \hphantom{-} 0 \\ \hphantom{-} 0 & \hphantom{-} 2 & \hphantom{-} 0 & -1 & \hphantom{-} 0 & \hphantom{-} 0 & \hphantom{-} 0 & \hphantom{-} 0 \\ -1 & \hphantom{-} 0 & \hphantom{-} 2 & -1 & \hphantom{-} 0 & \hphantom{-} 0 & \hphantom{-} 0 & \hphantom{-} 0 \\ \hphantom{-} 0 & -1 & -1 & \hphantom{-} 2 & -1 & \hphantom{-} 0 & \hphantom{-} 0 & \hphantom{-} 0 \\ \hphantom{-} 0 & \hphantom{-} 0 & \hphantom{-} 0 & -1 & \hphantom{-} 2 & -1 & \hphantom{-} 0 & \hphantom{-} 0 \\ \hphantom{-} 0 & \hphantom{-} 0 & \hphantom{-} 0 & \hphantom{-} 0 & -1 & \hphantom{-} 2 & -1 & \hphantom{-} 0 \\ \hphantom{-} 0 & \hphantom{-} 0 & \hphantom{-} 0 & \hphantom{-} 0 & \hphantom{-} 0 & -1 & \hphantom{-} 2 & -1 \\ \hphantom{-} 0 & \hphantom{-} 0 & \hphantom{-} 0 & \hphantom{-} 0 & \hphantom{-} 0 & \hphantom{-} 0 & -1 & \hphantom{-} 2 \end{pmatrix}.
\end{equation*}
\notag
$$
This raises the following question: can one express the characteristic numbers, as well as the intersection ring of a flag manifold $G/P$, merely in terms of the Cartan matrix of the Lie group $G$? In this section we fulfill this expectation. 4.2. A numerical construction of a Weyl groupTherefore, let $C=(c_{i,j})_{n\times n}$ be the Cartan matrix of some compact simple Lie group $G$, and let $\mathbb{R}^{n}$ be the $n$-dimensional real vector space with basis $\{\omega_1,\dots,\omega_n\}$. Define endomorphisms $\sigma_{i}\in \operatorname{End}(\mathbb{R}^{n})$, $1\leqslant i\leqslant n$, in terms of $C$ by the formula
$$
\begin{equation*}
\sigma_i(\omega_k) =\begin{cases} \omega_k & \text{if } i\neq k;\\ \omega_k-(c_{k,1}\omega_1+\dotsb+c_{k,n}\omega_n) & \text{if } i= k. \end{cases}
\end{equation*}
\notag
$$
By the general properties of Cartan matrices we have $\sigma_{i}^{2}=\mathrm{id}$, implying that $\sigma_{i}\in \operatorname{Aut}(\mathbb{R}^{n})$. Further, the following can be shown. Lemma 4.1. The subgroup of $\operatorname{Aut}(R^{n})$ generated by the $\sigma_{i}$s is isomorphic to the Weyl group $W(G)$ of $G$. For each subset $K\subset \{1,\dots,n\}$ there is a parabolic subgroup $P=P_{K}$, unique up to conjugations on $G$, whose Weyl group $W(P)$ is generated by those $\sigma_{{j}}\in W(G)$ with $j\notin K$. Resorting to the length function $l$ on $W(G)$, we can furthermore embed the set $W(P;G)$ as a subset of the group $W(G)$:
$$
\begin{equation*}
W(P;G)=\{w\in W(G)\mid l(w_1)\geqslant l(w), \;w_1\in wW(P)\}
\end{equation*}
\notag
$$
(see [4]) and put $W^{m}(P;G):=\{w\in W(P;G)\mid l(w)=m\}$. By Lemma 4.1 each element $w\in W^{m}(P;G)$ admits a factorization of the form
$$
\begin{equation*}
w=\sigma_{{i}_1}\circ\cdots\circ\sigma_{i_m} \quad\text{with } 1\leqslant i_1,\dots,i_m\leqslant n,
\end{equation*}
\notag
$$
hence it can be denoted by $w=\sigma_{I}$, where $I=(i_1,\dots,i_m)$. Such expressions of $w$ may not be unique, but the ambiguity can be dispelled by employing the following notion. Furnish the set with the lexicographic order $\preceq$ on the multi-indices $I$s. We call a decomposition $w=\sigma_{I}$ minimized if $I\in D(w)$ is the minimal multi-index. Clearly, we have the following (see, for example, [9]). It follows that the set $W^{m}(P;G)$ is also ordered by the lexicographic order $\preceq$ on the multi-indices $I$, hence can uniquely be presented as
$$
\begin{equation}
W^{m}(P;G) =\{w_{m,i}\mid 1\leqslant i\leqslant\beta(m)\}, \qquad \beta(m):=|W^{m}(P;G)|,
\end{equation}
\tag{4.1}
$$
where $w_{m,i}$ denotes the $i$th element in $W^{m}(P;G)$. In [25] the package Decomposition in Mathematica is presented, whose function is stated below. Algorithm I: Decomposition. Input: the Cartan matrix $C=(a_{ij})_{n\times n}$ of $G$ and a subset $K\subset \{1,\dots,n\}$ to specify a parabolic subgroup $P$. Output: the set $W(P;G)$ specified by the minimized decompositions of its elements, together with the index system (4.1) imposed by the order $\preceq$. Example 4.3. Let $G=\operatorname{SU}(n)$ be the special unitary group, and let $k\in \{1,\dots, n- 1\}$. The flag manifold $G/P_{\{k\}}$ is the Grassmannian manifold $G_{n,k}$. Applying Decomposition to the case $G_{9,4}$ we obtain the following table of minimized decompositions (see Table 2) for elements $w\in W(P_{4};\operatorname{SU}(9))$ with $l(w)\leqslant 8$ arranged in the order imposed by (4.1). For more examples of the results produced by Decomposition we refer to [24], §§ 1.1–7.1. Table 2.
Geometrically, for any $w\in W(P;G)$ the Schubert variety $X_{w}$ can be constructed explicitly in terms of its minimized decomposition [4], [20]. 4.3. A unified formula for Schubert characteristicsGiven an element $w\in W(P_{K};G)$ with minimized decomposition
$$
\begin{equation*}
w=\sigma_{i_1}\circ \sigma_{i_2}\circ \cdots \circ \sigma_{i_{m}},\quad 1\leqslant i_1,i_2,\dots,i_{m}\leqslant n,
\end{equation*}
\notag
$$
the structure matrix of $w$ is the strictly upper triangular matrix $A_w=(a_{s,t})_{m\times m}$ defined by the Cartan matrix $C=(c_{i,j})_{n\times n}$ of $G$ as follows:
$$
\begin{equation*}
a_{s,t} =\begin{cases} 0 & \text{if } s\geqslant t,\\ -c_{i_{s},i_{t}} & \text{if } s<t. \end{cases}
\end{equation*}
\notag
$$
For example, recall that the Cartan matrix of the exceptional Lie group $G_{2}$ is
$$
\begin{equation*}
C=\begin{pmatrix} \hphantom{-} 2 & -1 \\ -3 & \hphantom{-} 2 \end{pmatrix}.
\end{equation*}
\notag
$$
By Lemma 4.1 the Weyl group $W(G_{2})$ has two generators, $\sigma_{1}$ and $\sigma_{2}$. Consider the following elements with length $4$:
$$
\begin{equation*}
u=\sigma_1\circ \sigma_2\circ \sigma_1\circ \sigma_2\quad\text{and} \quad v=\sigma_2\circ \sigma_1\circ \sigma_2\circ \sigma_1.
\end{equation*}
\notag
$$
From the Cartan matrix $C$ one obtains
$$
\begin{equation*}
A_{u}=\begin{pmatrix} 0 & \hphantom{-} 1 & -2 & \hphantom{-} 1 \\ 0 & \hphantom{-} 0 & \hphantom{-} 3 & -2 \\ 0 & \hphantom{-} 0 & \hphantom{-} 0 & \hphantom{-} 1 \\ 0 & \hphantom{-} 0 & \hphantom{-} 0 & \hphantom{-} 0 \end{pmatrix} \quad\text{and}\quad A_{v}=\begin{pmatrix} 0 & \hphantom{-} 3 & -2 & \hphantom{-} 3 \\ 0 & \hphantom{-} 0 & \hphantom{-} 1 & -2 \\ 0 & \hphantom{-} 0 & \hphantom{-} 0 & \hphantom{-} 3 \\ 0 & \hphantom{-} 0 & \hphantom{-} 0 & \hphantom{-} 0 \end{pmatrix}.
\end{equation*}
\notag
$$
Let $\mathbb{Z}[x_1,\dots,x_m]$ be the ring of polynomials in $x_1,\dots,x_m$ that is graded by $\deg x_{i}=1$, and let $\mathbb{Z}[x_1,\dots,x_m]^{(m)}$ be its subgroup generated by the monomials of degree $m$. Given a strictly upper triangular integer $m\times m$ matrix $A=(a_{{i,j}})$, the triangular operator $T_A$ associated to $A$ is the linear map
$$
\begin{equation*}
T_{A}\colon\mathbb{Z}[x_1,\dots,x_m]^{(m)}\to \mathbb{Z}
\end{equation*}
\notag
$$
defined recursively by the following elimination rules:
$$
\begin{equation*}
h =\sum_{0\leqslant r\leqslant m}h_r\cdot x_m^r \quad\text{with } h_r\in\mathbb{Z}[x_1,\dots,x_{m-1}]^{(m-r)},
\end{equation*}
\notag
$$
the operator $T_A$ is well defined by rules 1)–3). We have the following. Lemma 4.4. For any polynomial $h\in \mathbb{Z}[x_1,\dots,x_m]^{(m)}$ the number $T_A(h)$ is a polynomial of degree $m$ in the entries of the matrix $A$. Extending the main results of [19], [20], and [28] we showed in [32], Theorem 2.4, the following formula, which expresses the Schubert characteristics of a flag manifold $G/P$ as polynomials in the Cartan numbers of the group $G$. Theorem 4.5. Let $w\in W(P;G)$ be an element with minimized decomposition $\sigma_{i_1}\circ \cdots \circ \sigma_{i_m}$ and structure matrix $A_w$. For any monomial $s_{u_1}\cdots s_{u_k}$ of total degree $m$ in the Schubert classes one has where for a multi-index $I=\{j_1,\dots,j_t\}$ we set $|I|:=t$, Since the matrix $A_w$ is constructed from the Cartan matrix of the group $G$ in term of the minimized decomposition of $w$, while the operator $T_{A_w}$ is evaluated easily using elimination rules 1)–3) stated above, formula (4.2) indicates an effective algorithm to evaluate the numbers $c_{u_1,\dots,u_k}^w$. Combining these ideas, the package Characteristics in Mathematica was developed (see, for example, [25]), whose function is described as follows. Algorithm II: Characteristics. Input: the Cartan matrix $C=(a_{ij})_{n\times n}$ of $G$ and a subset $K\subset \{1,\dots,n\}$ to specify a parabolic subgroup $P$. Output: the characteristics $c_{u_1,\dots,u_k}^w$ of $G/P$. Example 4.6 (the characteristics of the Schubert monomials at the top degree). Let $G/P$ be a flag manifold with $\dim_{\mathbb C}G/P=m$. According to the basis theorem (Theorem 3.2) there exists a unique element $w_0\in W(P;G)$ such that $l(w_0)=m$ and the Schubert class $s_{w_0}$ generates the top degree cohomology $H^{2m}(G/P)=\mathbb{Z}$. It follows that, for any monomial $s_{u_1}\cdots s_{u_k}$ of total degree $m$ in the Schubert classes, the characteristic number $c_{u_1,\dots,u_k}^w$ is given by the equality The cohomology of the Grassmannians $G_{n,k}$ are the most classical and archetypal examples of intersection theory (see [59], [60], and [34], p. 4). Traditionally, the characteristics $c_{u_{1},u_{2}}^{w}$ are given by the combinatorial Littlewood–Richardson rule [52], rather than by a closed formula. In contrast, our formula (4.2) is practical for numerical computation. In accordance with the convention above we set
$$
\begin{equation*}
c_{r} :=s_{\{k-r+1,k-r+2,\dots,k\}}\in H^{2r}(G_{n,k}), \qquad r=1,\dots,k.
\end{equation*}
\notag
$$
Then $c_{r}$ is also the $r$th Chern class of the canonical $k$-dimensional complex vector bundles on $G_{n,k}$ [55]. Applying Characteristics to the case of $G_{9,4}$ we obtain the following table of characteristics for monomials of top degree $\dim_{\mathbb C} G_{9,4}=20$ in the Chern classes (see Table 3). Table 3.The characteristics of the Grassmaniann $G_{9,4}$
Characteristics works equally well for other types of flag manifolds. For example, consider the flag manifold $E_{6}/P_{\{2\}}$, where $P_{\{2\}}=S^{1}\cdot \operatorname{SU}(6)$. Following Bourbaki’s numbering of simple roots [9], let $y_{1}$, $y_{3}$, $y_{4}$, and $y_{6}$ be, respectively, the Schubert classes $s_{I}$ with Then the cohomology $H^*(E_6/P_{\{2\}})$ is generated by $y_{1}$, $y_{3}$, $y_{4}$, and $y_{6}$ by Theorem 3 in [29]. Applying Characteristics we obtain the following table of characteristics for all monomials of top degree $\dim_{\mathbb C} E_{6}/P_{\{2\}}=21$ in the Schubert generators $y_{i}$ (see Table 4).Table 4.The characteristics of the flag manifold $E_6/S^{1}\cdot\operatorname{SU}(6)$
The contents of Tables 3 and 4 are compatible with Schubert’s computation in Table 1. Let $M$ be the variety of complete conics in 3-space. Then $\dim_{\mathbb C} M=8$, while the ring $H^*(M)$ is generated by the Schubert’s symbols $\mu,\rho,\nu\in H^{2}(M)$ [23]. That is, the equalities in Table 1 consist of the characteristics of the monomials $\mu^{m}\nu ^{n}\rho ^{8-m-n}$ of top degree $8$ in the symbols $\mu$, $\rho$, and $\nu$. Remark 4.7. Geometrically, the operator $T_{A_w}$ in (4.2) handles integration along the Schubert cell $X_w$ (see [4] and [20]). Formula (4.2) was extended in [21] to compute the products of basis elements of Grothendieck’s $K$-theory of flag manifolds and in [26] to evaluate the Steenrod operations on Schubert classes. 4.4. The intersection rings of flag manifoldsAs in Example 4.6, let $c_{i}\in H^{2i}(G_{{n,k}})$ be the $i$th Chern class. Borel [6] showed that
$$
\begin{equation}
H^*(G_{n,k})=\mathbb{Z}[c_1,\dots,c_k] / \langle c_{n-k+1}^{-1},\dots,c_n^{-1}\rangle,
\end{equation}
\tag{4.3}
$$
where $c_j^{-1}$ denotes the component of the formal inverse of $1+c_1+\cdots+c_k$ in degree $j$, and where $\langle\dots\rangle$ denotes the ideal generated by the indicated polynomials. Comparing formula (4.3) with the contents of Table 3 reveals the following phenomena: the characteristic numbers are essential for enumerative geometry, but fail to be a concise way to characterize the structure of the ring $H^*(G_{n,k})$. It is the Weil problem that has motivated the following extension of Borel’s formula (4.3) to all flag manifolds. Theorem 4.8. For each flag manifold $G/P$ there exist Schubert classes $y_{1},\dots,y_{n}$ such that where $f_{i}\in \mathbb{Z}[y_{1},\dots,y_{n}]$, $1\leqslant i\leqslant m$, and where the numbers $n$ and $m$ are the minimum integers appearing in such a presentation. Proof. Let $H^+(G/P)$ be the subring of the cohomology $H^*(G/P)$ spanned by the homogeneous elements of positive degree, and let $D(H^*(G/P))$ be the ideal of decomposable elements of the ring $H^+(G/P)$. Since the cohomology $H^*(G/P)$ is torsion free and has a basis consisting of Schubert classes, there exist Schubert classes $y_1,\dots,y_n$ on $G/P$ that correspond to a basis of the quotient group $H^+(G/P)/D(H^*(G/P))$. It follows that the inclusion $\{y_1,\dots,y_n\}\subset H^*(G/P)$ induces a ring epimorphism By Hilbert’s basis theorem there exist finitely many polynomials such that $\ker\pi=\langle f_1,\dots,f_m\rangle$. Of course, we can assume that the number $m$ is minimum with respect to formula (4.4).
As the cardinality of a basis of the quotient group $H^+(G/P)/D(H^*(G/P))$, the number $n$ is an invariant of $G/P$. In addition, if one changes the generators $y_1,\dots,y_n$ to $y_1',\dots,y_n'$, then each old generator $y_i$ can be expressed as a polynomial $g_i$ in the new ones $y_1',\dots,y_n'$. The invariance of $m$ is shown by the presentation where $f_j'$ is obtained from $f_j$ by substituting the $g_i$ for the $y_i$, $1\leqslant j\leqslant m$.The proof of Theorem 4.8 singles out two crucial steps in resolving the Weil problem:
Algorithm III: Chow-ring. Input: the Cartan matrix $C=(a_{ij})_{n\times n}$ of $G$, and a subset $K\subset \{1,\dots,n\}$ to specify a parabolic subgroup $P$. Output: a presentation (4.4) of the cohomology $H^*(G/P)$. Example 4.9. If $G$ is a simple Lie group of rank $n$ and if $K=\{1,\dots,n\}$, then the parabolic subgroup $P_K$ is a maximal torus $T$ in $G$, and the flag manifold $G/T$ is called the complete flag manifold of the group $G$. As applications of Chow-ring, the cohomologies $H^*(G/T)$ for the exceptional Lie groups have been determined in terms of a minimal system of generators and relations in Schubert classes (see, for example, [31]). We present below the results for $G=F_4,E_6,E_7$. (i) $H^*(F_4/T)=\mathbb{Z}[\omega_1,\dots,\omega_4,y_3,y_4] / \langle\rho_2,\rho_4,r_3,r_6,r_8,r_{12}\rangle$, where
$$
\begin{equation*}
\begin{aligned} \, \rho_2 & =c_2-4\omega_1^2,\\ \rho_4 & =3y_4+2\omega_1y_3-c_4,\\ r_3 & =2y_3-\omega_1^3,\\ r_6 & =y_3^2+2c_6-3\omega_1^2y_4,\\ r_8 & =3y_4^2-\omega_1^2c_6,\\ r_{12} & =y_4^3-c_6^2; \end{aligned}
\end{equation*}
\notag
$$
(ii) $H^*(E_6/T)=\mathbb{Z}[\omega_1,\dots,\omega_6,y_3,y_4] / \langle\rho_2,\rho_3,\rho_4,\rho_5,r_6,r_8,r_9,r_{12}\rangle$, where
$$
\begin{equation*}
\begin{aligned} \, \rho_2 & =4\omega_2^2-c_2,\\ \rho_3 & =2y_3+2\omega_2^3-c_3,\\ \rho_4 & =3y_4+\omega_2^4-c_4,\\ \rho_5 & =2\omega_2^2y_3-\omega_2c_4+c_5,\\ r_6 & =y_3^2-\omega_2c_5+2c_6,\\ r_8 &=3y_4^2-2c_5y_3-\omega_2^2c_6+\omega_2^3c_5,\\ r_9 & =2y_3c_6-\omega_2^3c_6,\\ r_{12} & =y_4^3-c_6^2; \end{aligned}
\end{equation*}
\notag
$$
(iii) $H^*(E_7/T)=\mathbb{Z}[\omega_1,\dots,\omega_7,y_3,y_4,y_5,y_9]/\langle \rho_i,r_j\rangle$, where
$$
\begin{equation*}
\begin{aligned} \, \rho_2 & =4\omega_2^2-c_2,\\ \rho_3 & =2y_3+2\omega_2^3-c_3,\\ \rho_4 & =3y_4+\omega_2^4-c_4,\\ \rho_5 & =2y_5-2\omega_2^2y_3+\omega_2c_4-c_5,\\ r_6 & =y_3^2-\omega_2c_5+2c_6,\\ r_8 & =3y_4^2+2y_3y_5-2y_3c_5+2\omega_2c_7-\omega_2^2c_6+\omega_2^3c_5,\\ r_9 & =2y_9+2y_4y_5-2y_3c_6-\omega_2^2c_7+\omega_2^3c_6,\\ r_{10} & =y_5^2-2y_3c_7+\omega_2^3c_7,\\ r_{12} & =y_4^3-4y_5c_7-c_6^2-2y_3y_9-2y_3y_4y_5 +2\omega_2y_5c_6+3\omega_2y_4c_7+c_5c_7,\\ r_{14} & =c_7^2-2y_5y_9+2y_3y_4c_7-\omega_2^3y_4c_7,\\ r_{18} & =y_9^2+2y_5c_6c_7-y_4c_7^2-2y_4y_5y_9+2y_3y_5^3-5\omega_2y_5^2c_7, \end{aligned}
\end{equation*}
\notag
$$
where the set $\{\omega_i,\, 1\leqslant i\leqslant \operatorname{rank} G\}$ is the Schubert basis of $H^2(G/T)$, which is also the set of fundamental dominant weights of the relevant group $G$ (see, for example, [22], Lemma 2.4), the $c_{r}$ are certain polynomials in $\omega_1,\dots,\omega_n$ defined in [32], (5.17), whose geometric implication will be made transparent in Example 4.11 below, and where, in terms of the Bourbaki numbering of simple roots [9], the $y_i$ are the Schubert classes $s_{I}$ on $G/T$ specified in the table below:
For more examples of the applications of Chow-ring to computations with partial flag manifolds $G/P$, we refer to [29], Theorems 1–7. 4.5. Schubert polynomialsIn classical enumerative geometry the Chern class $c_i$ of the Grassmannian $G_{n,k}$ arises firstly as the Poincaré dual of the variety of $k$-planes meeting a general $(n-k-i)$-plane in the $n$-space $\mathbb{C}^{n}$ and is well known as the $i$th special Schubert class of $G_{n,k}$. The celebrated Giambelli formula [40] (1902), expressing an arbitrary Schubert class $s_w$ on $G_{n,k}$ as a determinant (that is, a polynomial) in the special ones, can be praised as the beginning of the idea of Schubert polynomials. In general, suppose that $G/P$ is a flag manifold for which a solution (4.4) to the Weil problem is available. Then every Schubert class $s_w$ can be expressed as a polynomial in the generators $y_1,\dots,y_n$. Inspired by Giambelli’s formula, we may call the generators $y_1,\dots,y_n$ the special Schubert classes of $G/P$ and ask the question of expressing an arbitrary Schubert class $s_w$ on $G/P$ as a polynomial $\mathcal{G}_w$ in the special ones, where $\deg \mathcal{G}_w=2l(w)$. In particular, starting from Borel’s presentation of the cohomology ring of the flag manifold $U(n)/T$
$$
\begin{equation*}
H^*(U(n)/T) =\mathbb{Z}[x_1,\dots,x_n]/\langle e_1,\dots,e_n\rangle
\end{equation*}
\notag
$$
(see [6]), where $e_1,\dots,e_n$ are the elementary symmetric polynomials in $x_1,\dots,x_n$, Lascoux and Schützenberger [50] proposed to solve the problem in accordance with the following rules:
In the traditional approach to Schubert polynomials, there are two necessary prerequisites (see, for instance, [5]):
Algorithm IV: Schubert polynomials. Input: a set $\{y_1,\dots,y_n\}$ of special Schubert classes on $G/P$ and an integer $m>0$. Output: the Schubert polynomials $\mathcal{G}_w$ for $w\in W^m(H;G)$. We clarify the details of Algorithm IV. Let $\mathbb{Z}[y_1,\dots,y_n]^{(m)}$ be a group of polynomials of degree $m$ in the special Schubert classes $y_1,\dots,y_n$. We examine the map
$$
\begin{equation*}
\pi_{m}\colon \mathbb{Z}[y_1,\dots,y_n]^{(m)}\to H^{2m}(G/P)
\end{equation*}
\notag
$$
induced by the inclusion $y_1,\dots,y_n\in H^{*}(G/T)$. Let be the monomial basis of the group $\mathbb{Z}[y_1,\dots,y_n]^{(m)}$, and recall (see (4.1)) that the Schubert basis of the group $H^{2m}(G/P)$ is $\{s_{m,1},\dots,s_{m,\beta(m)}\}$. Since each $y^{\alpha}\in B(m)$ is a monomial in the special Schubert classes, Algorithm II is applicable to expand it linearly in $\{s_{m,1},\dots,s_{m,\beta(m)}\}$ to get a $b(m)\times \beta(m)$ matrix $M(\pi_m)$ which satisfies the linear system
$$
\begin{equation*}
\begin{pmatrix} y^{\alpha_1} \\ \vdots \\ y^{\alpha_{{b(m)}}} \end{pmatrix} =M(\pi_m) \begin{pmatrix} s_{{m,1}} \\ \vdots \\ s_{{m,\beta (m)}} \end{pmatrix}.
\end{equation*}
\notag
$$
Moreover, since the map $\pi_m$ is surjective, the matrix $M(\pi_m)$ has a $\beta(m)\times \beta(m)$ minor equal to $\pm 1$. Thus, the standard integral row and column operation diagonalizing $M(\pi_m)$ (see [57], pp. 162–164) provides us with two invertible matrices $P=P_{b(m)\times b(m)}$ and $Q=Q_{{\beta (m)\times \beta (m)}}$ satisfying the relation
$$
\begin{equation}
PM(\pi_m)Q= \begin{pmatrix} I_{{\beta (m)}} \\ C \end{pmatrix}_{{b(m)\times \beta (m)}},
\end{equation}
\tag{4.5}
$$
where $I_{\beta(m)}$ denotes the identity matrix of rank $\beta(m)$. Summarizing, Algorithm IV can be realized by the following procedure. Step 1. Compute $M(\pi_m)$ using Characteristics. Step 2. Diagonalize $M(\pi_m)$ to obtain the matrices $P$ and $Q$ in (4.5). Step 3. Set
$$
\begin{equation*}
\begin{pmatrix} \mathcal{G}_{m,1} \\ \vdots \\ \mathcal{G}_{m,\beta(m)} \end{pmatrix} :=Q\cdot[P] \begin{pmatrix} y^{\alpha_1} \\ \vdots \\ y^{\alpha_{b(m)}} \end{pmatrix},
\end{equation*}
\notag
$$
where $[P]$ is the $\beta(m)\times\beta(m)$ matrix formed by the first $\beta(m)$ rows of $P$. The following is clear. Theorem 4.10. The polynomial $\mathcal{G}_{m,k}(y_1,\dots,y_n)$ is a Schubert polynomial of the Schubert class $s_{m,k}$, $1\leqslant k\leqslant \beta(m)$. Example 4.11. Algorithm IV is suitable for those flag manifolds $G/P$ whose top degree Schubert polynomials are not known (or are in question). For the exceptional Lie group $G=E_{n}$ with $n=6,7,8$ the parabolic subgroup $P_{\{2\}}$ has a canonical $n$-dimensional complex representation, which gives rise to the canonical complex $n$-bundle $\xi_n$ on the flag manifold $E_n/P_{\{2\}}$ (see [1]). According to Borel and Hirzebruch (see [8], § 10), the Chern classes $c_i(\xi_n)$ can be expressed as a (rather lengthy) polynomial in the positive roots of the group $E_n$. However, in the notation for special Schubert classes of $E_n/P_{\{2\}}$ specified in the table
using Algorithm IV we obtain the following concise expressions of the Chern classes $c_i(\xi_n)$ as polynomials in the special Schubert classes (see Table 5). Table 5.
In addition, the Schubert polynomials of the flag manifold $E_{6}/P_{\{2\}}$ in degrees $m=8,9$ are listed in Table 6. Table 6.
Remark 4.12. Presumably, Schubert polynomials can be useful to compute the characteristics. However, in our approach Schubert characteristics are a preliminary step toward Schubert polynomials (see, for example, Algorithm IV). Currently, the theory of Schubert polynomials is a powerful tool for discovering the combinatorial structure of the Littlewood–Richardson coefficients [16], [11], [12], and it is essential for the geometric topic of the degeneracy loci of maps between vector bundles [39], [66]. There have also been extensive studies of Schubert polynomials in quantum cohomology [35] and in the $K$-theory of flag manifolds. For recent progress in this branch of contemporary Schubert calculus, we refer to the articles by Kirillov and Narus [46], and Smirnov and Tutubalina [64]. 4.6. Applications to the topology of homogeneous spacesFor a compact Lie group $G$ with a closed subgroup $H$ the quotient space $G/H$ is called a homogeneous space of $G$. In contrast to flag manifolds, the cohomology of a homogeneous space can be non-trivial in odd degrees, and can contain torsion elements. A classical problem of topology is to express the cohomology of a Lie group $G$ or a homogeneous space $G/H$ by a minimal system of explicit generators and relations. The traditional approaches due to H. Cartan, A. Borel, Baum, and Toda utilize various spectral sequence techniques [3], [6], [44], [67], [71], and the calculation encounters the same difficulties when applied to a Lie group $G$ whose integral cohomology has torsion elements, in particular, when $G$ is one of the exceptional Lie groups. Schubert calculus makes the cohomology theory of homogeneous spaces appear in a new light. For example, after substituting formulae (i)–(iii) from Example 4.9 into the second page of the Serre spectral sequence of the fibration $G\to G/T$, the integral cohomology $H^*(G)$, as well as the Hopf algebra structure on the $\mod p$ cohomology $H^*(G;\mathbb{Z}_p)$, have been determined by computations with the Schubert classes on $G/T$ (see [27] and [30]). For more examples of the extension of Schubert calculus to computations with homogeneous spaces, see [29], § 5.5. Concluding remarksThroughout ages, a common hope of geometers was to find calculable mechanisms among the geometric entities they are dealing with (for example, algebraic varieties, cellular complexes, vector bundles, or the cobordism classes of smooth manifolds). The emergence of Schubert calculus, or the birth of intersection theory, catered to this demand. Today Schubert calculus has been widely integrated into many branches of mathematics, and has profoundly affected the trajectories of the development of such fields as the theory of characteristic classes [55], string theory [45], and algebraic combinatorics [37]. All this has vigorously witnessed Hilbert’s broad vision and foresight, and at the same time, has put forward the essential request to explore effective rules for performing computations. Subject to the plan, this article has recalled the earlier studies on Schubert calculus, presented a solution of the problem of characteristics, and illustrated a passage from the Cartan matrices of Lie groups to the intersection theory of flag manifolds, in which the characteristics play a central role. For the historic significance and rigorous treatment of the enumerative examples of Schubert mentioned in § 3, we refer to the survey articles by Kleiman [47], [49], or the relevant sections of the books by Fulton [38] and Eisenbud and Harris [34] on intersection theory. For other computer systems that can be used to perform certain computations in the intersection rings of flag manifolds, see, for example, Nikolenko and Semenov [56] (the package ChowMaple06), Grayson et al. [41], [42] (the package Schubert2 in Macaulay2), and Decker et al. [17] (the library Schubert in Singular). The authors would like to thank their referees for improvements over the earlier version of this paper.
Citation in format AMSBIB
\Bibitem{DuaZha22} |
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