View source for Schubert cell

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]]).

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

<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>

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>

<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>

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" />.

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====
<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>