View source for Schubert variety

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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|Grassmann manifold]] $G_{n,m}$. Schubert varieties define a basis of the
[[Chow ring|Chow ring]] $A(G_{n,m})$, and for $k=\C$ &mdash; 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
{{Cite|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 ({{Cite|Bo}}). The geometry of Schubert varieties was studied, e.g., in
{{Cite|De}},
{{Cite|LaSe}}.

====References====
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|valign="top"|{{Ref|Bo}}||valign="top"| A. Borel, "Linear algebraic groups", Benjamin (1969) pp. 283ff {{MR|0251042}} {{ZBL|0206.49801}} {{ZBL|0186.33201}}
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|valign="top"|{{Ref|De}}||valign="top"| M. Demazure, "Désingularisation des variétés de Schubert généralisées" ''Ann. Sci. Ecole Norm. Sup.'', '''7''' (1974) pp. 53–87 {{MR|0354697}} {{ZBL|0312.14009}}
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|valign="top"|{{Ref|GrHa}}||valign="top"| P.A. Griffiths, J.E. Harris, "Principles of algebraic geometry", '''1''', Wiley (Interscience) (1978) {{MR|0507725}} {{ZBL|0408.14001}}
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|valign="top"|{{Ref|HoPe}}||valign="top"| W.V.D. Hodge, D. Pedoe, "Methods of algebraic geometry", '''2''', Cambridge Univ. Press (1954) {{MR|0061846}} {{ZBL|0055.38705}}
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|valign="top"|{{Ref|Kl}}||valign="top"| S.L. Kleiman, "Problem 15. Rigorous foundation of Schubert's enumerative calculus" F.E. Browder (ed.), ''Mathematical developments arising from Hilbert problems'', ''Proc. Symp. Pure Math.'', '''28''', Amer. Math. Soc. (1976) pp. 445–482 {{MR|0429938}} {{ZBL|}}
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|valign="top"|{{Ref|LaSe}}||valign="top"| V. Lakshmibai, C. Seshadri, "Geometry of $G/P$  V." ''J. of Algebra'', '''100''' (1986) pp. 462–557 {{MR|840589}} {{ZBL|0618.14026 }}
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|valign="top"|{{Ref|Sc}}||valign="top"| H. Schubert, "Lösung des Charakteristiken-Problems für lineare Räume beliebiger Dimension" ''Mitt. Math. Gesellschaft Hamburg'', '''1''' (1889) pp. 134–155 {{MR|}} {{ZBL|18.0631.01}}
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