Difference between revisions of "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 $k=\C$ — 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 | |
| − | The notion of a Schubert variety has been generalized to any complete homogeneous space of a semi-simple linear algebraic group | + | {{Cite|De}}, |
| + | {{Cite|LaSe}}. | ||
====References==== | ====References==== | ||
| − | + | {| | |
| + | |- | ||
| + | |valign="top"|{{Ref|Bo}}||valign="top"| A. Borel, "Linear algebraic groups", Benjamin (1969) pp. 283ff {{MR|0251042}} {{ZBL|0206.49801}} {{ZBL|0186.33201}} | ||
| + | |- | ||
| + | |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}} | ||
| + | |- | ||
| + | |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}} | ||
| + | |- | ||
| + | |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}} | ||
| + | |- | ||
| + | |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|}} | ||
| + | |- | ||
| + | |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 }} | ||
| + | |- | ||
| + | |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}} | ||
| + | |- | ||
| + | |} | ||
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
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