# Schubert variety

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2010 Mathematics Subject Classification: Primary: 14M15 [MSN][ZBL]

A Schubert variety is the set of all \$m\$-dimensional subspaces \$W\$ of an \$n\$-dimensional vector space \$V\$ over a field \$k\$ satisfying the Schubert conditions: \$\dim(W\cap V_j) \ge j\$, \$j=1,\dots,m\$, where \$V_1\subset\cdots\subset V_m\$ is a fixed flag of subspaces of \$V\$. In Grassmann coordinates these conditions are given by linear equations; a Schubert variety is an irreducible (generally speaking, singular) algebraic subvariety of the Grassmann manifold \$G_{n,m}\$. Schubert varieties define a basis of the Chow ring \$A(G_{n,m})\$, and for \$k=\C\$ — a basis for the homology group \$H_*(G_{n,m},\Z)\$.

The Schubert conditions were considered by H. Schubert in connection with enumeration problems for geometric objects with given incidence properties. Hilbert's 15th problem concerns a foundation for the enumeration theory developed by Schubert (see [Kl]).

The notion of a Schubert variety has been generalized to any complete homogeneous space of a semi-simple linear algebraic group \$G\$. It is the Zariski closure of any Bruhat cell ([Bo]). The geometry of Schubert varieties was studied, e.g., in [De], [LaSe].

How to Cite This Entry:
Schubert variety. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Schubert_variety&oldid=25380
This article was adapted from an original article by A.L. Onishchik (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article