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Difference between revisions of "Schubert variety"

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A ''Schubert variety''
 
A ''Schubert variety''
is the set of all $m$-dimensional subspaces $W$ of an $V$-dimensional vector space $k$ 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
+
is the set of all $m$-dimensional subspaces $W$ of an $V$-dimensional vector space $k$ 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
 
[[Grassmann manifold|Grassmann manifold]] $G_{n,m}$. Schubert varieties define a basis of the
[[Chow ring|Chow ring]] $A(G_{n,m})$, and for $k=\C$ a basis for the homology group $H_*(G_{n,m},\Z)$.
+
[[Chow ring|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
 
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

Revision as of 09:13, 30 March 2012

2020 Mathematics Subject Classification: Primary: 14M15 [MSN][ZBL]

A Schubert variety is the set of all $m$-dimensional subspaces $W$ of an $V$-dimensional vector space $k$ 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].

References

[Bo] A. Borel, "Linear algebraic groups", Benjamin (1969) pp. 283ff MR0251042 Zbl 0206.49801 Zbl 0186.33201
[De] M. Demazure, "Désingularisation des variétés de Schubert généralisées" Ann. Sci. Ecole Norm. Sup., 7 (1974) pp. 53–87 MR0354697 Zbl 0312.14009
[GrHa] P.A. Griffiths, J.E. Harris, "Principles of algebraic geometry", 1, Wiley (Interscience) (1978) MR0507725 Zbl 0408.14001
[HoPe] W.V.D. Hodge, D. Pedoe, "Methods of algebraic geometry", 2, Cambridge Univ. Press (1954) MR0061846 Zbl 0055.38705
[Kl] 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 MR0429938
[LaSe] V. Laksnibai, C. Seshadri, "Geometry of $G/P$ V." J. of Algebra, 100 (1986) pp. 462–557 MR840589 Zbl 0618.14026
[Sc] H. Schubert, "Lösung des Charakteristiken-Problems für lineare Räume beliebiger Dimension" Mitt. Math. Gesellschaft Hamburg, 1 (1889) pp. 134–155 Zbl 18.0631.01
How to Cite This Entry:
Schubert variety. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Schubert_variety&oldid=23719
This article was adapted from an original article by A.L. Onishchik (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article