Schubert cell

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