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The set of all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083430/s0834301.png" />-dimensional subspaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083430/s0834302.png" /> of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083430/s0834303.png" />-dimensional vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083430/s0834304.png" /> over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083430/s0834305.png" /> satisfying the Schubert conditions: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083430/s0834306.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083430/s0834307.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083430/s0834308.png" /> is a fixed flag of subspaces of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083430/s0834309.png" />. 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]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083430/s08343010.png" />. Schubert varieties define a basis of the [[Chow ring|Chow ring]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083430/s08343011.png" />, and for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083430/s08343012.png" /> — a basis for the homology group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083430/s08343013.png" />.
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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 [[#References|[2]]]).
 
 
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD 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}} </TD></TR><TR><TD valign="top">[2]</TD> <TD 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|}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> P.A. Griffiths, J.E. Harris, "Principles of algebraic geometry" , '''1''' , Wiley (Interscience) (1978) {{MR|0507725}} {{ZBL|0408.14001}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> W.V.D. Hodge, D. Pedoe, "Methods of algebraic geometry" , '''2''' , Cambridge Univ. Press (1954) {{MR|0061846}} {{ZBL|0055.38705}} </TD></TR></table>
 
  
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A ''Schubert variety''
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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
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[[Grassmann manifold|Grassmann manifold]] $G_{n,m}$. Schubert varieties define a basis of the
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[[Chow ring|Chow ring]] $A(G_{n,m})$, and for $k=\C$ &mdash; a basis for the homology group $H_*(G_{n,m},\Z)$.
  
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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
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{{Cite|Kl}}).
  
====Comments====
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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
The notion of a Schubert variety has been generalized to any complete homogeneous space of a semi-simple linear algebraic group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083430/s08343014.png" />. It is the Zariski closure of any Bruhat cell ([[#References|[a1]]]). The geometry of Schubert varieties was studied, e.g., in [[#References|[a2]]], [[#References|[a3]]].
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{{Cite|De}},
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{{Cite|LaSe}}.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> A. Borel, "Linear algebraic groups" , Benjamin (1969) pp. 283ff {{MR|0251042}} {{ZBL|0206.49801}} {{ZBL|0186.33201}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> M. Demazure, "Désingularisation des variétés de Schubert généralisés" ''Ann. Sci. Ecole Norm. Sup.'' , '''7''' (1974) pp. 53–87 {{MR|}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> V. Laksnibai, C. Seshadri, "Geometry of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083430/s08343015.png" /> - V" ''J. of Algebra'' , '''100''' (1986) pp. 462–557 {{MR|840589}} {{ZBL|}} </TD></TR></table>
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{|
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|-
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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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|-
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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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|-
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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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|-
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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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|-
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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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|-
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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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|-
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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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|-
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|}

Latest revision as of 16:26, 9 December 2023

2020 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].

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. Lakshmibai, 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=21931
This article was adapted from an original article by A.L. Onishchik (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article