View source for Schubert calculus

''Schubert enumerative calculus''

A formal calculus of symbols representing geometric conditions used to solve problems in enumerative geometry. This originated in work of M. Chasles [[#References|[a3]]] on conics and was systematized and used to great effect by H. Schubert in [[#References|[a13]]]. 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 [[#References|[a7]]] for a complete reference on intersection theory; for historical surveys and a discussion of enumerative geometry, see [[#References|[a9]]], [[#References|[a10]]].

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 [[#References|[a12]]]. For example, how many lines in projective <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130080/s1300801.png" />-space meet <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130080/s1300802.png" /> 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 [[#References|[a8]]], linear algebra [[#References|[a6]]] and systems theory [[#References|[a2]]].

The Grassmannian <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130080/s1300803.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130080/s1300804.png" />-dimensional subspaces (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130080/s1300805.png" />-planes) in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130080/s1300806.png" /> over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130080/s1300807.png" /> has distinguished Schubert varieties

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130080/s1300808.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130080/s1300809.png" /> is a [[Flag|flag]] of linear subspaces with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130080/s13008010.png" />. The [[Schubert cycle|Schubert cycle]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130080/s13008011.png" /> is the cohomology class Poincaré dual to the fundamental homology cycle of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130080/s13008012.png" /> (cf. also [[Homology|Homology]]). The basis theorem asserts that the Schubert cycles form a basis of the Chow ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130080/s13008013.png" /> (when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130080/s13008014.png" /> is the complex number field, these are the integral cohomology groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130080/s13008015.png" />) of the Grassmannian with

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130080/s13008016.png" /></td> </tr></table>

(see also [[Grassmann manifold|Grassmann manifold]]). The duality theorem asserts that the basis of Schubert cycles is self-dual under the intersection pairing

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130080/s13008017.png" /></td> </tr></table>

with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130080/s13008018.png" /> dual to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130080/s13008019.png" />.

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130080/s13008020.png" /> be a special Schubert cycle (cf. [[Schubert cycle|Schubert cycle]]). Then

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130080/s13008021.png" /></td> </tr></table>

the sum running over all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130080/s13008022.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130080/s13008023.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130080/s13008024.png" />. 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130080/s13008025.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130080/s13008026.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130080/s13008027.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130080/s13008028.png" />. Then Giambelli's formula is

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130080/s13008029.png" /></td> </tr></table>

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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130080/s13008030.png" />-planes meeting <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130080/s13008031.png" /> general <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130080/s13008032.png" />-planes non-trivially is the coefficient of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130080/s13008033.png" /> in the product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130080/s13008034.png" />, which is [[#References|[a14]]]

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130080/s13008035.png" /></td> </tr></table>

These four results hold more generally for cohomology rings of flag manifolds <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130080/s13008036.png" />; Schubert cycles form a self-dual basis, the Chevalley formula [[#References|[a4]]] determines the ring structure (when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130080/s13008037.png" /> is a [[Borel subgroup|Borel subgroup]]), and the Bernshtein–Gel'fand–Gel'fand formula [[#References|[a1]]] and Demazure formula [[#References|[a5]]] 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 [[#References|[a11]]] 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130080/s13008038.png" /> corresponds to the partition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130080/s13008039.png" />; 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130080/s13008040.png" />.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  I.N. Bernshtein,   I.M. Gel'fand,   S.I. Gel'fand,   "Schubert cells and cohomology of the spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130080/s13008041.png" />"  ''Russian Math. Surveys'' , '''28''' :  3  (1973)  pp. 1–26</TD></TR><TR><TD valign="top">[a2]</TD> <TD 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</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  M. Chasles,   "Construction des coniques qui satisfont à cinque conditions"  ''C.R. Acad. Sci. Paris'' , '''58'''  (1864)  pp. 297–308</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  C. Chevalley,   "Sur les décompositions cellulaires des espaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130080/s13008042.png" />"  W. Haboush (ed.) , ''Algebraic Groups and their Generalizations: Classical Methods'' , ''Proc. Symp. Pure Math.'' , '''56:1''' , Amer. Math. Soc.  (1994)  pp. 1–23</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  M. Demazure,   "Désingularization des variétés de Schubert généralisées"  ''Ann. Sci. École Norm. Sup. (4)'' , '''7'''  (1974)  pp. 53–88</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  W. Fulton,   "Eigenvalues, invariant factors, highest weights, and Schubert calculus"  ''Bull. Amer. Math. Soc.'' , '''37'''  (2000)  pp. 209–249</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  W. Fulton,   "Intersection theory" , ''Ergebn. Math.'' , '''2''' , Springer  (1998)  (Edition: Second)</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  B. Huber,   F. Sottile,   B. Sturmfels,   "Numerical Schubert calculus"  ''J. Symbolic Comput.'' , '''26''' :  6  (1998)  pp. 767–788</TD></TR><TR><TD valign="top">[a9]</TD> <TD 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</TD></TR><TR><TD valign="top">[a10]</TD> <TD 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</TD></TR><TR><TD valign="top">[a11]</TD> <TD valign="top">  D.E. Littlewood,   A.R. Richardson,   "Group characters and algebra"  ''Philos. Trans. Royal Soc. London.'' , '''233'''  (1934)  pp. 99–141</TD></TR><TR><TD valign="top">[a12]</TD> <TD valign="top">  S.L. Kleiman,   D. Laksov,   "Schubert calculus"  ''Amer. Math. Monthly'' , '''79'''  (1972)  pp. 1061–1082</TD></TR><TR><TD valign="top">[a13]</TD> <TD valign="top">  H. Schubert,   "Kälkul der abzählenden Geometrie" , Springer  (1879)  (Reprinted (with an introduction by S. Kleiman): 1979)</TD></TR><TR><TD valign="top">[a14]</TD> <TD valign="top">  H. Schubert,   "Anzahl-Bestimmungen für lineare Räume beliebiger Dimension"  ''Acta Math.'' , '''8'''  (1886)  pp. 97–118</TD></TR></table>