Schubert cell
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
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where
![]() |
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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) MR1102012 Zbl 0726.20030 |
Schubert cell. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Schubert_cell&oldid=23716