In mathematics, Schubert polynomials are generalizations of
Schur polynomials
In mathematics, Schur polynomials, named after Issai Schur, are certain symmetric polynomials in ''n'' variables, indexed by partitions, that generalize the elementary symmetric polynomials and the complete homogeneous symmetric polynomials. In r ...
that represent cohomology classes of
Schubert cycle In algebraic geometry, a Schubert variety is a certain subvariety of a Grassmannian, usually with singular points. Like a Grassmannian, it is a kind of moduli space, whose points correspond to certain kinds of subspaces ''V'', specified using li ...
s in
flag varieties In mathematics, a generalized flag variety (or simply flag variety) is a homogeneous space whose points are flag (linear algebra), flags in a finite-dimensional vector space ''V'' over a field (mathematics), field F. When F is the real or complex nu ...
. They were introduced by and are named after
Hermann Schubert
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Hermann Cäsar Hannibal Schubert (22 May 1848 – 20 July 1911) 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 nu ...
.
Background
described the history of Schubert polynomials.
The Schubert polynomials
are polynomials in the variables
depending on an element
of the infinite symmetric group
of all permutations of
fixing all but a finite number of elements. They form a basis for the polynomial ring