Schubert calculus

2020 Mathematics Subject Classification: Primary: 14M15 [MSN][ZBL]

The Schubert calculus or Schubert enumerative calculus is a formal calculus of symbols representing geometric conditions used to solve problems in enumerative geometry. This originated in work of M. Chasles [Ch] on conics and was systematized and used to great effect by H. Schubert in [Sc]. The justification of Schubert's enumerative calculus and the verification of the numbers he obtained was the contents of Hilbert's 15th problem (cf. also Hilbert problems).

Justifying Schubert's enumerative calculus was a major theme of twentieth century algebraic geometry, and intersection theory provides a satisfactory modern framework. Enumerative geometry deals with the second part of Hilbert's problem. See [Fu2] for a complete reference on intersection theory; for historical surveys and a discussion of enumerative geometry, see [Kl], [Kl2].

The Schubert calculus also refers to mathematics arising from the following class of enumerative geometric problems: Determine the number of linear subspaces of projective space that satisfy incidence conditions imposed by other linear subspaces. For a survey, see [KlLa]. For example, how many lines in projective $3$-space meet $4$ given lines? These problems are solved by studying both the geometry and the cohomology or Chow rings of Grassmann varieties (cf. also Chow ring; Grassmann manifold). This field of Schubert calculus enjoys important connections not only to algebraic geometry and algebraic topology, but also to algebraic combinatorics, representation theory, differential geometry, linear algebraic groups, and symbolic computation, and has found applications in numerical homotopy continuation [HuSoSt], linear algebra [Fu] and systems theory [By].

The Grassmannian $G_{m,n}$ of $m$-dimensional subspaces ($m$-planes) in $\def\P{\mathbb{P}}\P^n$ over a field $k$ has distinguished Schubert varieties

$$\def\O{\Omega}\def\a{\alpha}\O_{a_0,\dots,a_m}V_*:= \{W\in G_{m,n} : W\cap V_{a_j}\ge j\},$$ where $V_*:V_0\subset\cdots\subset V_n=\P^n$ is a flag of linear subspaces with $\dim V_j = j$. The Schubert cycle $\def\s{\sigma}\s_{a_0,\dots,a_n}$ is the cohomology class Poincaré dual to the fundamental homology cycle of $\O_{a_0,\dots,a_m} V_*$ (cf. also Homology). The basis theorem asserts that the Schubert cycles form a basis of the Chow ring $A^* G_{m,n}$ (when $k$ is the complex number field, these are the integral cohomology groups $H^* G_{m,n}$) of the Grassmannian with

$$\s_{a_0,\dots,a_m}\in A^{(m+1)(n+1)-{m+1\choose n+1} -a_0-\cdots -a_m} G_{m.n},$$ (see also Grassmann manifold). The duality theorem asserts that the basis of Schubert cycles is self-dual under the intersection pairing

$$\def\b{\beta} (\a,\b) \in H^* G_{m,n} \otimes H^* G_{m,n} \to\deg(\a \cdot \b) = \int_{G_{m,n}} \a\cdot\b$$ with $\s_{a_0,\dots,a_m}$ dual to $\s_{n-a_m,\dots,n-a_0}$.

Let $\def\t{\tau}\t_b := \s_{n-m-b,n-m+1,\dots,n}$ be a special Schubert cycle (cf. Schubert cycle). Then

$$\s_{a_0,\dots,a_m}\cdot \t_b = \sum \s_{c_0,\dots,c_m},$$ the sum running over all $(c_0,\dots,c_m)$ with $0\le c_0\le a_0\le c_1\le a_1\cdots\le c_m\le a_m$ and $b = \sum_i(a_i-c_i)$. This Pieri formula determines the ring structure of cohomology; an algebraic consequence is the Giambelli formula for expressing an arbitrary Schubert cycle in terms of special Schubert cycles. Define $\t_b = 0$ if $B<0$ or $B>m$, and $\t_0 = 1$. Then Giambelli's formula is

$$\s_{a_0,\dots,a_m} = \det(\t_{n-m+j-a_i})_{i,j=0,\dots,m}.$$ These four results enable computation in the Chow ring of the Grassmannian, and the solution of many problems in enumerative geometry. For instance, the number of $m$-planes meeting $(m+1)(n-m)$ general $(n-m-1)$-planes non-trivially is the coefficient of $\s_{0,\dots,m}$ in the product $(\t_1)^{(m+1)(n-m)}$, which is [Sc2] $$\frac{1!\cdots(n-m-1)!\cdots ((m+1)(n-m))!}{(n-m)!(n-m+1)!\cdots(n!-1)!}.$$ These four results hold more generally for cohomology rings of flag manifolds $G/P$; Schubert cycles form a self-dual basis, the Chevalley formula [Ch2] determines the ring structure (when $P$ is a Borel subgroup), and the Bernshtein–Gel'fand–Gel'fand formula [BeGeGe] and Demazure formula [De] give the analogue of the Giambelli formula. More explicit Giambelli formulas are provided by Schubert polynomials.

One cornerstone of the Schubert calculus for the Grassmannian is the Littlewood–Richardson rule [LiRi] for expressing a product of Schubert cycles in terms of the basis of Schubert cycles. (This rule is usually expressed in terms of an alternative indexing of Schubert cycles using partitions. A sequence $(a_0,\dots,a_m)$ corresponds to the partition $(n-m-a_0,n-m+1,\dots,n-a_m)$; cf. Schur functions in algebraic combinatorics.) The analogue of the Littlewood–Richardson rule is not known for most other flag varieties $G/P$.

References

[BeGeGe]	I.N. Bernshtein, I.M. Gel'fand, S.I. Gel'fand, "Schubert cells and cohomology of the spaces $G/P$" Russian Math. Surveys, 28 : 3 (1973) pp. 1–26 MR0686277
[By]	C.I. Byrnes, "Algebraic and geometric aspects of the control of linear systems" C.I. Byrnes (ed.) C.F. Martin (ed.), Geometric Methods in Linear systems Theory, Reidel (1980) pp. 85–124
[Ch]	M. Chasles, "Construction des coniques qui satisfont à cinque conditions" C.R. Acad. Sci. Paris, 58 (1864) pp. 297–308
[Ch2]	C. Chevalley, "Sur les décompositions cellulaires des espaces $G/B$" W. Haboush (ed.), Algebraic Groups and their Generalizations: Classical Methods, Proc. Symp. Pure Math., 56:1, Amer. Math. Soc. (1994) pp. 1–23 MR1278698 Zbl 0824.14042
[De]	M. Demazure, "Désingularization des variétés de Schubert généralisées" Ann. Sci. École Norm. Sup. (4), 7 (1974) pp. 53–88
[Fu]	W. Fulton, "Eigenvalues, invariant factors, highest weights, and Schubert calculus" Bull. Amer. Math. Soc., 37 (2000) pp. 209–249 MR1754641 Zbl 0994.15021
[Fu2]	W. Fulton, "Intersection theory", Ergebn. Math., 2, Springer (1998) (Edition: Second) MR1644323 Zbl 0885.14002
[HuSoSt]	B. Huber, F. Sottile, B. Sturmfels, "Numerical Schubert calculus" J. Symbolic Comput., 26 : 6 (1998) pp. 767–788 MR1662035 Zbl 1064.14508
[Kl]	S. Kleiman, "Problem 15: Rigorous foundation of Schubert's enumerative calculus", Mathematical Developments arising from Hilbert Problems, Proc. Symp. Pure Math., 28, Amer. Math. Soc. (1976) pp. 445–482 MR429938
[Kl2]	S. Kleiman, "Intersection theory and enumerative geometry: A decade in review" S. Bloch (ed.), Algebraic Geometry (Bowdoin, 1985), Proc. Symp. Pure Math., 46:2, Amer. Math. Soc. (1987) pp. 321–370 MR0927987 Zbl 0664.14031
[KlLa]	S.L. Kleiman, D. Laksov, "Schubert calculus" Amer. Math. Monthly, 79 (1972) pp. 1061–1082 MR0323796 Zbl 0272.14016
[LiRi]	D.E. Littlewood, A.R. Richardson, "Group characters and algebra" Philos. Trans. Royal Soc. London., 233 (1934) pp. 99–141 Zbl 0009.20203 Zbl 60.0896.01
[Sc]	H. Schubert, "Kalkül der abzählenden Geometrie", Springer (1879) (Reprinted (with an introduction by S. Kleiman): 1979) MR0555576
[Sc2]	H. Schubert, "Anzahl-Bestimmungen für lineare Räume beliebiger Dimension" Acta Math., 8 (1886) pp. 97–118 Zbl 18.0632.01