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The orbit of a [[Borel subgroup|Borel subgroup]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130090/s1300901.png" /> on a flag variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130090/s1300902.png" /> [[#References|[a1]]], 14.12. Here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130090/s1300903.png" /> is a semi-simple [[Linear algebraic group|linear algebraic group]] over an [[Algebraically closed field|algebraically closed field]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130090/s1300904.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130090/s1300905.png" /> is a [[Parabolic subgroup|parabolic subgroup]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130090/s1300906.png" /> so that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130090/s1300907.png" /> is a complete homogeneous variety. Schubert cells are indexed by the cosets of the [[Weyl group|Weyl group]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130090/s1300908.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130090/s1300909.png" /> in the Weyl group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130090/s13009010.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130090/s13009011.png" />. Choosing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130090/s13009012.png" />, these cosets are identified with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130090/s13009013.png" />-fixed points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130090/s13009014.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130090/s13009015.png" /> is a [[Maximal torus|maximal torus]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130090/s13009016.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130090/s13009017.png" />. The fixed points are conjugates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130090/s13009018.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130090/s13009019.png" /> containing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130090/s13009020.png" />. The orbit <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130090/s13009021.png" />, the [[Affine space|affine space]] of dimension equal to the length of the shortest element of the coset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130090/s13009022.png" />. When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130090/s13009023.png" /> is the complex number field, Schubert cells constitute a CW-decomposition of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130090/s13009024.png" /> (cf. also [[CW-complex|CW-complex]]).
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{{MSC|14M15}}
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{{TEX|done}}
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130090/s13009025.png" /> be any [[Field|field]] and suppose <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130090/s13009026.png" /> is the Grassmannian <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130090/s13009027.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130090/s13009028.png" />-planes in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130090/s13009029.png" /> (cf. also [[Grassmann manifold|Grassmann manifold]]). Schubert cells for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130090/s13009030.png" /> arise in an elementary manner. Among the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130090/s13009031.png" /> by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130090/s13009032.png" /> matrices whose row space is a given <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130090/s13009033.png" />, there is a unique echelon matrix
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A ''Schubert cell'' is
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the orbit of a
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[[Borel subgroup|Borel subgroup]] $B\subset G$ on a flag variety $G/P$
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{{Cite|Bo}}, 14.12. Here, $G$ is a semi-simple
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[[Linear algebraic group|linear algebraic group]] over an
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[[Algebraically closed field|algebraically closed field]] $k$ and $P$ is a
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[[Parabolic subgroup|parabolic subgroup]] of $G$ so that $G/P$ is a complete homogeneous variety. Schubert cells are indexed by the cosets of the
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[[Weyl group|Weyl group]] $W_P$ of $P$ in the Weyl group $W$ of $G$. Choosing $B\subset P$, these cosets are identified with $T$-fixed points of $G/P$, where $T$ is a
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[[Maximal torus|maximal torus]] of $G$ and $T\subset B$. The fixed points are conjugates $P'$ of $P$ containing $T$. The orbit $BwW_P\simeq \mathbb{A}^{l(wW_P)}$, the
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[[Affine space|affine space]] of dimension equal to the length of the shortest element of the coset $wW_P$. When $k$ is the complex number field, Schubert cells constitute a CW-decomposition of $G/P$ (cf. also
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[[CW-complex|CW-complex]]).
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130090/s13009034.png" /></td> </tr></table>
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Let $k$ be any
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[[Field|field]] and suppose $G/P$ is the Grassmannian $G_{m,n}$ of $m$-planes in $k^n$ (cf. also
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[[Grassmann manifold|Grassmann manifold]]). Schubert cells for $G_{m,n}$ arise in an elementary manner. Among the $m$ by $n$ matrices whose row space is a given $H\in G_{m,n}$, there is a unique echelon matrix
  
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$$(E_0 \ E_1\ E_2\ \dots\ E_n)$$
 
where
 
where
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130090/s13009035.png" /></td> </tr></table>
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$$E_0 = \begin{pmatrix}* &\dots& *\\ \vdots & \ddots & \vdots\\ * &\dots& *\end{pmatrix}, E_1 = \begin{pmatrix}1&0&\dots&0\\
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0&* &\dots& *\\ \vdots&\vdots & \ddots & \vdots\\ 0&* &\dots& *\end{pmatrix}, $$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130090/s13009036.png" /></td> </tr></table>
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$$E_2 = \begin{pmatrix}0&0&\dots&0\\1&0&\dots&0\\0&* &\dots& *\\ \vdots&\vdots & \ddots & \vdots\\ 0&* &\dots& *\end{pmatrix}, \dots,
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E_n = \begin{pmatrix}0&0&\dots&0\\\vdots&\vdots&\dots&\vdots\\0&0&\dots&0\\1&*&\dots&*\end{pmatrix},$$
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where $*$ represents an arbitrary element of $k$.
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130090/s13009037.png" /> represents an arbitrary element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130090/s13009038.png" />.
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This echelon representative of $H$ is computed from any representative by Gaussian elimination (cf. also
 
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[[Elimination theory|Elimination theory]]). The column numbers $a_1<\dots<a_m$ of the leading entries ($1$s) of the rows in this echelon representative determine the type of $H$. Counting the undetermined entries in such an echelon matrix shows that the set of $H\in G_{m,n}$ with this type is isomorphic to $\mathbb{A}^{mn-\sum(a_i+i-1)}$. This set is a Schubert cell of $G_{m,n}$.
This echelon representative of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130090/s13009039.png" /> is computed from any representative by Gaussian elimination (cf. also [[Elimination theory|Elimination theory]]). The column numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130090/s13009040.png" /> of the leading entries (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130090/s13009041.png" />s) of the rows in this echelon representative determine the type of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130090/s13009042.png" />. Counting the undetermined entries in such an echelon matrix shows that the set of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130090/s13009043.png" /> with this type is isomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130090/s13009044.png" />. This set is a Schubert cell of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130090/s13009045.png" />.
 
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> A. Borel, "Linear algebraic groups" , ''Grad. Texts Math.'' , '''126''' , Springer (1991) (Edition: Second) {{MR|1102012}} {{ZBL|0726.20030}} </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", ''Grad. Texts Math.'', '''126''', Springer (1991) (Edition: Second) {{MR|1102012}} {{ZBL|0726.20030}}
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|-
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|}

Latest revision as of 18:38, 30 March 2012

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

A Schubert cell is the orbit of a Borel subgroup $B\subset G$ on a flag variety $G/P$ [Bo], 14.12. Here, $G$ is a semi-simple linear algebraic group over an algebraically closed field $k$ and $P$ is a parabolic subgroup of $G$ so that $G/P$ is a complete homogeneous variety. Schubert cells are indexed by the cosets of the Weyl group $W_P$ of $P$ in the Weyl group $W$ of $G$. Choosing $B\subset P$, these cosets are identified with $T$-fixed points of $G/P$, where $T$ is a maximal torus of $G$ and $T\subset B$. The fixed points are conjugates $P'$ of $P$ containing $T$. The orbit $BwW_P\simeq \mathbb{A}^{l(wW_P)}$, the affine space of dimension equal to the length of the shortest element of the coset $wW_P$. When $k$ is the complex number field, Schubert cells constitute a CW-decomposition of $G/P$ (cf. also CW-complex).

Let $k$ be any field and suppose $G/P$ is the Grassmannian $G_{m,n}$ of $m$-planes in $k^n$ (cf. also Grassmann manifold). Schubert cells for $G_{m,n}$ arise in an elementary manner. Among the $m$ by $n$ matrices whose row space is a given $H\in G_{m,n}$, there is a unique echelon matrix

$$(E_0 \ E_1\ E_2\ \dots\ E_n)$$ where

$$E_0 = \begin{pmatrix}* &\dots& *\\ \vdots & \ddots & \vdots\\ * &\dots& *\end{pmatrix}, E_1 = \begin{pmatrix}1&0&\dots&0\\ 0&* &\dots& *\\ \vdots&\vdots & \ddots & \vdots\\ 0&* &\dots& *\end{pmatrix}, $$

$$E_2 = \begin{pmatrix}0&0&\dots&0\\1&0&\dots&0\\0&* &\dots& *\\ \vdots&\vdots & \ddots & \vdots\\ 0&* &\dots& *\end{pmatrix}, \dots, E_n = \begin{pmatrix}0&0&\dots&0\\\vdots&\vdots&\dots&\vdots\\0&0&\dots&0\\1&*&\dots&*\end{pmatrix},$$ where $*$ represents an arbitrary element of $k$.

This echelon representative of $H$ is computed from any representative by Gaussian elimination (cf. also Elimination theory). The column numbers $a_1<\dots<a_m$ of the leading entries ($1$s) of the rows in this echelon representative determine the type of $H$. Counting the undetermined entries in such an echelon matrix shows that the set of $H\in G_{m,n}$ with this type is isomorphic to $\mathbb{A}^{mn-\sum(a_i+i-1)}$. This set is a Schubert cell of $G_{m,n}$.

References

[Bo] A. Borel, "Linear algebraic groups", Grad. Texts Math., 126, Springer (1991) (Edition: Second) MR1102012 Zbl 0726.20030
How to Cite This Entry:
Schubert cell. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Schubert_cell&oldid=21929
This article was adapted from an original article by Frank Sottile (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article