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Schubert cell

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The orbit of a Borel subgroup on a flag variety [a1], 14.12. Here, is a semi-simple linear algebraic group over an algebraically closed field and is a parabolic subgroup of so that is a complete homogeneous variety. Schubert cells are indexed by the cosets of the Weyl group of in the Weyl group of . Choosing , these cosets are identified with -fixed points of , where is a maximal torus of and . The fixed points are conjugates of containing . The orbit , the affine space of dimension equal to the length of the shortest element of the coset . When is the complex number field, Schubert cells constitute a CW-decomposition of (cf. also CW-complex).

Let be any field and suppose is the Grassmannian of -planes in (cf. also Grassmann manifold). Schubert cells for arise in an elementary manner. Among the by matrices whose row space is a given , there is a unique echelon matrix

where

where represents an arbitrary element of .

This echelon representative of is computed from any representative by Gaussian elimination (cf. also Elimination theory). The column numbers of the leading entries (s) of the rows in this echelon representative determine the type of . Counting the undetermined entries in such an echelon matrix shows that the set of with this type is isomorphic to . This set is a Schubert cell of .

References

[a1] A. Borel, "Linear algebraic groups" , Grad. Texts Math. , 126 , Springer (1991) (Edition: Second)
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