Schubert calculus
Encyclopedia
In mathematics
, Schubert calculus is a branch of algebraic geometry
introduced in the nineteenth century by Hermann Schubert
, in order to solve various counting problems of projective geometry
(part of enumerative geometry
). It was a precursor of several more modern theories, for example characteristic class
es, and in particular its algorithmic aspects are still of current interest.
The objects introduced by Schubert are the Schubert cells, 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 details see Schubert variety
.
The intersection theory of these cells, which can be seen as the product structure in the cohomology ring
of the Grassmannian of associated cohomology classes, in principle allows the prediction of the cases where intersections of cells results in a finite set of points; which are potentially concrete answers to enumerative questions. A supporting theoretical result is that the Schubert cells (or rather, their classes) span the whole cohomology ring.
In detailed calculations the combinatorial aspects enter as soon as the cells have to be indexed. Lifted from the Grassmanian, 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 (by block matrix
).
Putting Schubert's system on a rigorous footing is Hilbert's 15th problem
.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, Schubert calculus is a branch of algebraic geometry
Algebraic geometry
Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...
introduced in the nineteenth century by Hermann Schubert
Hermann Schubert
Hermann Cäsar Hannibal Schubert was a German mathematician.Schubert was one of the leading developers of enumerative geometry, which considers those parts of algebraic geometry that involve a finite number of solutions. In 1874, Schubert won a prize for solving a question posed by Zeuthen...
, in order to solve various counting problems of projective geometry
Projective geometry
In mathematics, projective geometry is the study of geometric properties that are invariant under projective transformations. This means that, compared to elementary geometry, projective geometry has a different setting, projective space, and a selective set of basic geometric concepts...
(part of enumerative geometry
Enumerative geometry
In mathematics, enumerative geometry is the branch of algebraic geometry concerned with counting numbers of solutions to geometric questions, mainly by means of intersection theory.-History:...
). It was a precursor of several more modern theories, for example characteristic class
Characteristic class
In mathematics, a characteristic class is a way of associating to each principal bundle on a topological space X a cohomology class of X. The cohomology class measures the extent to which the bundle is "twisted" — particularly, whether it possesses sections or not...
es, and in particular its algorithmic aspects are still of current interest.
The objects introduced by Schubert are the Schubert cells, which are locally closed sets in a Grassmannian
Grassmannian
In mathematics, a Grassmannian is a space which parameterizes all linear subspaces of a vector space V of a given dimension. For example, the Grassmannian Gr is the space of lines through the origin in V, so it is the same as the projective space P. The Grassmanians are compact, topological...
defined by conditions of incidence
Incidence (geometry)
In geometry, the relations of incidence are those such as 'lies on' between points and lines , and 'intersects' . That is, they are the binary relations describing how subsets meet...
of a linear subspace in projective space with a given flag
Flag manifold
In mathematics, a generalized flag variety is a homogeneous space whose points are flags in a finite-dimensional vector space V over a field F. When F is the real or complex numbers, a generalized flag variety is a smooth or complex manifold, called a real or complex flag manifold...
. For details see Schubert variety
Schubert variety
In algebraic geometry, a Schubert variety is a certain subvariety of a Grassmannian, usually with singular points. Described by means of linear algebra, a typical example consists of the k-dimensional subspaces V of an n dimensional vector space W, such that\dim\ge jfor j = 1, 2, ..., k,...
.
The intersection theory of these cells, which can be seen as the product structure in the cohomology ring
Cohomology ring
In mathematics, specifically algebraic topology, the cohomology ring of a topological space X is a ring formed from the cohomology groups of X together with the cup product serving as the ring multiplication. Here 'cohomology' is usually understood as singular cohomology, but the ring structure is...
of the Grassmannian of associated cohomology classes, in principle allows the prediction of the cases where intersections of cells results in a finite set of points; which are potentially concrete answers to enumerative questions. A supporting theoretical result is that the Schubert cells (or rather, their classes) span the whole cohomology ring.
In detailed calculations the combinatorial aspects enter as soon as the cells have to be indexed. Lifted from the Grassmanian, which is a homogeneous space
Homogeneous space
In mathematics, particularly in the theories of Lie groups, algebraic groups and topological groups, a homogeneous space for a group G is a non-empty manifold or topological space X on which G acts continuously by symmetry in a transitive way. A special case of this is when the topological group,...
, to the general linear group
General linear group
In mathematics, the general linear group of degree n is the set of n×n invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again invertible, and the inverse of an invertible matrix is invertible...
that acts on it, similar questions are involved in the Bruhat decomposition
Bruhat decomposition
In mathematics, the Bruhat decomposition G = BWB into cells can be regarded as a general expression of the principle of Gauss–Jordan elimination, which generically writes a matrix as a product of an upper triangular and lower triangular matrices—but with exceptional cases...
and classification of parabolic subgroups (by block matrix
Block matrix
In the mathematical discipline of matrix theory, a block matrix or a partitioned matrix is a matrix broken into sections called blocks. Looking at it another way, the matrix is written in terms of smaller matrices. We group the rows and columns into adjacent 'bunches'. A partition is the rectangle...
).
Putting Schubert's system on a rigorous footing is Hilbert's 15th problem
Hilbert's problems
Hilbert's problems form a list of twenty-three problems in mathematics published by German mathematician David Hilbert in 1900. The problems were all unsolved at the time, and several of them were very influential for 20th century mathematics...
.