In mathematics, Schubert calculus^[1] is a branch of algebraic geometry introduced in the nineteenth century by Hermann Schubert in order to solve various counting problems of projective geometry and, as such, is viewed as part of enumerative geometry. Giving it a more rigorous foundation was the aim of Hilbert's 15th problem. It is related to several more modern concepts, such as characteristic classes, and both its algorithmic aspects and applications remain of current interest. The term Schubert calculus is sometimes used to mean the enumerative geometry of linear subspaces of a vector space, which is roughly equivalent to describing the cohomology ring of Grassmannians. Sometimes it is used to mean the more general enumerative geometry of algebraic varieties that are homogenous spaces of simple Lie groups. Even more generally, Schubert calculus is sometimes understood as encompassing the study of analogous questions in generalized cohomology theories.

The objects introduced by Schubert are the Schubert cells,^[2] which are locally closed sets in a Grassmannian defined by conditions of incidence of a linear subspace in projective space with a given flag. For further details see Schubert variety.

The intersection theory^[3] of these cells, which can be seen as the product structure in the cohomology ring of the Grassmannian, consisting of associated cohomology classes, allows in particular the determination of cases in which the intersections of cells results in a finite set of points. A key result is that the Schubert cells (or rather, the classes of their Zariski closures, the Schubert cycles or Schubert varieties) span the whole cohomology ring.

The combinatorial aspects mainly arise in relation to computing intersections of Schubert cycles. Lifted from the Grassmannian, which is a homogeneous space, to the general linear group that acts on it, similar questions are involved in the Bruhat decomposition and classification of parabolic subgroups (as block triangular matrices).

Schubert calculus can be constructed using the Chow ring ^[3] of the Grassmannian, where the generating cycles are represented by geometrically defined data.^[4] Denote the Grassmannian of ${\displaystyle k}$-planes in a fixed ${\displaystyle n}$-dimensional vector space ${\displaystyle V}$ as ${\displaystyle \mathbf {Gr} (k,V)}$, and its Chow ring as ${\displaystyle A^{*}(\mathbf {Gr} (k,V))}$. (Note that the Grassmannian is sometimes denoted ${\displaystyle \mathbf {Gr} (k,n)}$ if the vector space isn't explicitly given or as ${\displaystyle \mathbb {G} (k-1,n-1)}$ if the ambient space ${\displaystyle V}$ and its ${\displaystyle k}$-dimensional subspaces are replaced by their projectizations.) Choosing an (arbitrary) complete flag

${\displaystyle {\mathcal {V}}=(V_{1}\subset \cdots \subset V_{n-1}\subset V_{n}=V),\quad \dim {V}_{i}=i,\quad i=1,\dots ,n,}$

to each weakly decreasing ${\displaystyle k}$-tuple of integers ${\displaystyle \mathbf {a} =(a_{1},\ldots ,a_{k})}$, where

${\displaystyle n-k\geq a_{1}\geq a_{2}\geq \cdots \geq a_{k}\geq 0,}$

i.e., to each partition of weight

${\displaystyle |\mathbf {a} |=\sum _{i=1}^{k}a_{i},}$

whose Young diagram fits into the ${\displaystyle k\times (n-k)}$ rectangular one for the partition ${\displaystyle (n-k)^{k}}$, we associate a Schubert variety^[1][2] (or Schubert cycle) ${\displaystyle \Sigma _{\mathbf {a} }({\mathcal {V}})\subset \mathbf {Gr} (k,V)}$, defined as

${\displaystyle \Sigma _{\mathbf {a} }({\mathcal {V}})=\{w\in \mathbf {Gr} (k,V):\dim(V_{n-k+i-a_{i}}\cap w)\geq i{\text{ for }}i=1,\dots ,k\}.}$

This is the closure, in the Zariski topology, of the Schubert cell^[1][2]

${\displaystyle X_{\mathbf {a} }({\mathcal {V}}):=\{w\in \mathbf {Gr} (k,V):\dim(V_{j}\cap w)=i{\text{ for all }}n-k-a_{i}+i\leq j\leq n-k-a_{i+1}+i,\quad 1\leq j\leq n\}\subset \Sigma _{\mathbf {a} }({\mathcal {V}}),}$

which is used when considering cellular homology instead of the Chow ring. The latter are disjoint affine spaces, of dimension ${\displaystyle |\mathbf {a} |}$, whose union is ${\displaystyle \mathbf {Gr} (k,V)}$.

An equivalent characterization of the Schubert cell ${\displaystyle X_{\mathbf {a} }({\mathcal {V}})}$ may be given in terms of the dual complete flag

${\displaystyle {\tilde {\mathcal {V}}}=({\tilde {V}}_{1}\subset {\tilde {V}}_{2}\cdots \subset {\tilde {V}}_{n}=V),}$

where

${\displaystyle {\tilde {V}}_{i}:=V_{n}\backslash V_{n-i},\quad i=1,\dots ,n\quad (V_{0}:=\emptyset ).}$

Then ${\displaystyle X_{\mathbf {a} }({\mathcal {V}})\subset \mathbf {Gr} (k,V)}$ consists of those ${\displaystyle k}$-dimensional subspaces ${\displaystyle w\subset V}$ that have a basis ${\displaystyle ({\tilde {W}}_{1},\dots ,{\tilde {W}}_{k})}$ consisting of elements

${\displaystyle {\tilde {W}}_{i}\in {\tilde {V}}_{k+a_{i}-i+1},\quad i=1,\dots ,k}$

of the subspaces ${\displaystyle \{{\tilde {V}}_{k+a_{i}-i+1}\}_{i=1,\dots ,k}.}$

Since the homology class ${\displaystyle [\Sigma _{\mathbf {a} }({\mathcal {V}})]\in A^{*}(\mathbf {Gr} (k,V))}$, called a Schubert class, does not depend on the choice of complete flag ${\displaystyle {\mathcal {V}}}$, it can be written as

${\displaystyle \sigma _{\mathbf {a} }:=[\Sigma _{\mathbf {a} }]\in A^{*}(\mathbf {Gr} (k,V)).}$

It can be shown that these classes are linearly independent and generate the Chow ring as their linear span. The associated intersection theory is called Schubert calculus. For a given sequence ${\displaystyle \mathbf {a} =(a_{1},\ldots ,a_{j},0,\ldots ,0)}$ with ${\displaystyle a_{j}>0}$ the Schubert class ${\displaystyle \sigma _{(a_{1},\ldots ,a_{j},0,\ldots ,0)}}$ is usually just denoted ${\displaystyle \sigma _{(a_{1},\ldots ,a_{j})}}$. The Schubert classes given by a single integer ${\displaystyle \sigma _{a_{1}}}$, (i.e., a horizontal partition), are called special classes. Using the Giambelli formula below, all the Schubert classes can be generated from these special classes.

There is an easy description of the cohomology ring, or the Chow ring, of the Grassmannian ${\displaystyle \mathbf {Gr} (k,V)}$ using the Chern classes of two natural vector bundles over ${\displaystyle \mathbf {Gr} (k,V)}$. We have the exact sequence of vector bundles over ${\displaystyle \mathbf {Gr} (k,V)}$

${\displaystyle 0\to T\to {\underline {V}}\to Q\to 0}$

where ${\displaystyle T}$ is the tautological bundle whose fiber, over any element ${\displaystyle w\in \mathbf {Gr} (k,V)}$ is the subspace ${\displaystyle w\subset V}$ itself, ${\displaystyle \,{\underline {V}}:=\mathbf {Gr} (k,V)\times V}$ is the trivial vector bundle of rank ${\displaystyle n}$, with ${\displaystyle V}$ as fiber and ${\displaystyle Q}$ is the quotient vector bundle of rank ${\displaystyle n-k}$, with ${\displaystyle V/w}$ as fiber. The Chern classes of the bundles ${\displaystyle T}$ and ${\displaystyle Q}$ are

${\displaystyle c_{i}(T)=(-1)^{i}\sigma _{(1)^{i}},}$

where ${\displaystyle (1)^{i}}$ is the partition whose Young diagram consists of a single column of length ${\displaystyle i}$ and

${\displaystyle c_{i}(Q)=\sigma _{i}.}$

The tautological sequence then gives the presentation of the Chow ring as

${\displaystyle A^{*}(\mathbf {Gr} (k,V))={\frac {\mathbb {Z} [c_{1}(T),\ldots ,c_{k}(T),c_{1}(Q),\ldots ,c_{n-k}(Q)]}{(c(T)c(Q)-1)}}.}$

One of the classical examples analyzed is the Grassmannian ${\displaystyle \mathbf {Gr} (2,4)}$ since it parameterizes lines in ${\displaystyle \mathbb {P} ^{3}}$. Using the Chow ring ${\displaystyle A^{*}(\mathbf {Gr} (2,4))}$, Schubert calculus can be used to compute the number of lines on a cubic surface.^[4]

• Enumerative geometry
• Chow ring
• Intersection theory
• Grassmannian
• Giambelli's formula
• Pieri's formula
• Chern class
• Quintic threefold
• Mirror symmetry conjecture