View source for Schubert calculus

{{MSC|14M15}}
{{TEX|done}}

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
{{Cite|Ch}} on conics and was systematized and used to great effect by H. Schubert in
{{Cite|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|Hilbert problems]]).

Justifying Schubert's enumerative calculus was a major theme of twentieth century
[[Algebraic geometry|algebraic geometry]], and
[[Intersection theory|intersection theory]] provides a satisfactory modern framework. Enumerative geometry deals with the second part of Hilbert's problem. See
{{Cite|Fu2}} for a complete reference on intersection theory; for historical surveys and a discussion of enumerative geometry, see
{{Cite|Kl}},
{{Cite|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
{{Cite|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|Chow ring]];
[[Grassmann manifold|Grassmann manifold]]). This field of Schubert calculus enjoys important connections not only to algebraic geometry and
[[Algebraic topology|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
{{Cite|HuSoSt}}, linear algebra
{{Cite|Fu}} and systems theory
{{Cite|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|flag]] of linear subspaces with $\dim V_j = j$. The
[[Schubert cycle|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|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|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|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
{{Cite|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
{{Cite|Ch2}} determines the ring structure (when $P$ is a
[[Borel subgroup|Borel subgroup]]), and the Bernshtein–Gel'fand–Gel'fand formula
{{Cite|BeGeGe}} and Demazure formula
{{Cite|De}} give the analogue of the Giambelli formula. More explicit Giambelli formulas are provided by
[[Schubert polynomials|Schubert polynomials]].

One cornerstone of the Schubert calculus for the Grassmannian is the Littlewood–Richardson rule
{{Cite|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|Schur functions in algebraic combinatorics]].) The analogue of the Littlewood–Richardson rule is not known for most other flag varieties $G/P$.

====References====
{|
|-
|valign="top"|{{Ref|BeGeGe}}||valign="top"| 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 {{MR|0686277}} {{ZBL|}}
|-
|valign="top"|{{Ref|By}}||valign="top"| 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 {{MR|}} {{ZBL|}}
|-
|valign="top"|{{Ref|Ch}}||valign="top"| M. Chasles, "Construction des coniques qui satisfont à cinque conditions" ''C.R. Acad. Sci. Paris'', '''58''' (1864) pp. 297–308 {{MR|}} {{ZBL|}}
|-
|valign="top"|{{Ref|Ch2}}||valign="top"| 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 {{MR|1278698}} {{ZBL|0824.14042}}
|-
|valign="top"|{{Ref|De}}||valign="top"| M. Demazure, "Désingularization des variétés de Schubert généralisées" ''Ann. Sci. École Norm. Sup. (4)'', '''7''' (1974) pp. 53–88 {{MR|}} {{ZBL|}}
|-
|valign="top"|{{Ref|Fu}}||valign="top"| W. Fulton, "Eigenvalues, invariant factors, highest weights, and Schubert calculus" ''Bull. Amer. Math. Soc.'', '''37''' (2000) pp. 209–249 {{MR|1754641}} {{ZBL|0994.15021}}
|-
|valign="top"|{{Ref|Fu2}}||valign="top"| W. Fulton, "Intersection theory", ''Ergebn. Math.'', '''2''', Springer (1998) (Edition: Second) {{MR|1644323}} {{ZBL|0885.14002}}
|-
|valign="top"|{{Ref|HuSoSt}}||valign="top"| B. Huber, F. Sottile, B. Sturmfels, "Numerical Schubert calculus" ''J. Symbolic Comput.'', '''26''' : 6 (1998) pp. 767–788 {{MR|1662035}} {{ZBL|1064.14508}}
|-
|valign="top"|{{Ref|Kl}}||valign="top"| 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 {{MR|429938}} {{ZBL|}}
|-
|valign="top"|{{Ref|Kl2}}||valign="top"| 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 {{MR|0927987}} {{ZBL|0664.14031}}
|-
|valign="top"|{{Ref|KlLa}}||valign="top"| S.L. Kleiman, D. Laksov, "Schubert calculus" ''Amer. Math. Monthly'', '''79''' (1972) pp. 1061–1082 {{MR|0323796}} {{ZBL|0272.14016}}
|-
|valign="top"|{{Ref|LiRi}}||valign="top"| D.E. Littlewood, A.R. Richardson, "Group characters and algebra" ''Philos. Trans. Royal Soc. London.'', '''233''' (1934) pp. 99–141 {{MR|}} {{ZBL|0009.20203}} {{ZBL|60.0896.01}}
|-
|valign="top"|{{Ref|Sc}}||valign="top"| H. Schubert, "Kalkül der abzählenden Geometrie", Springer (1879) (Reprinted (with an introduction by S. Kleiman): 1979) {{MR|0555576}} {{ZBL|}}
|-
|valign="top"|{{Ref|Sc2}}||valign="top"| H. Schubert, "Anzahl-Bestimmungen für lineare Räume beliebiger Dimension" ''Acta Math.'', '''8''' (1886) pp. 97–118 {{MR|}} {{ZBL|18.0632.01}}
|-
|}