Let $SL(n)$ be algebraic group defined over finite field $\mathbb{F}_{p^n}$, $B$ be Borel subgroup consist of upper triangular matrices and $T$ be maximal torus consist of diagonal matrices. Let $W$ be Weyl group, defined by $N_{G}T/T$. Let $G/B=\cup_{w\in W} B\cdot e_{w}$ be the Bruhat decomposition, where $e_{w}=wB$. Denote $X:=B\cdot e_{w}$ be Schubert cell and $O^{-}$ be opposite big cell i.e.$B^{-}\cdot e_{id}$, where $B^{-}$ is the opposite Borel.

My question is how to calculate the number of $\mathbb{F}_{p^n}$-points of $Y:=X\cap O^{-}$. Is this one polynomial count?(i.e. there exists a polynomial $f$ with $\mathbb{Z}$-coefficients not depend on $n$, such that the number of $\mathbb{F}_{p^n}$-points is $f(p^n)$)


1 Answer 1


Yes, this is polynomial point count. An intersection of a Bruhat cell and an opposite Bruhat cell is called a "Richardson variety", and Richardson varieties come with decompositions known as Deodhar decompositions. Each piece of the Deodhar decomposition is of the form $\mathbb{G}_m^{N-2k} \times \mathbb{A}^k$ where $N$ is the dimension of the Richardson, so it has $(q-1)^{N-2k} q^k$ points over $\mathbb{F}_q$, and adding them all up gives a polynomial in $q$.

To give the first nontrivial example, in $\text{GL}_3$, the big Schubert cell is flags which can be written as $$\begin{bmatrix} 1&0&0 \\ x&1&0 \\ z&y&1 \\ \end{bmatrix} B.$$ The intersection with the opposite cell is the open locus $z(xy-z) \neq 0$. We can decompose this as the union of the pieces where $x\neq 0$ and where $x = 0$, so we get $$\left\{ \begin{bmatrix} 1&0&0 \\ x&1&0 \\ z&\tfrac{w+z}{x}&1 \\ \end{bmatrix} B : wxz \neq 0 \right\} \sqcup \left\{ \begin{bmatrix} 1&0&0 \\ 0&1&0 \\ z&y&1 \\ \end{bmatrix} B : z \neq 0 \right\},$$ so we have $$(q-1)^3 + (q-1)q$$ points.

The Deodhar stratification of $(B_+ w B_+ \cap B_- u B_+)/B_+$ depends on a choice of reduced word for $w$. Deodhar's original paper is

Deodhar, Vinay V., On some geometric aspects of Bruhat orderings. I: A finer decomposition of Bruhat cells, Invent. Math. 79, 499-511 (1985). ZBL0563.14023.

For very explicit formulas, you might also like

Marsh, R. J.; Rietsch, K., Parametrizations of flag varieties, Represent. Theory 8, 212-242 (2004). ZBL1053.14057.

In your particular case, you want $w = w_0$ and $u = e$. If you take the reduced word $(s_1 s_2 \cdots s_{n-1}) (s_1 s_2 \cdots s_{n-2}) \cdots (s_1 s_2) (s_1)$, Allen Knutson pointed out to me that the Deodhar stratification has a very explicit form -- it corresponds to taking a lower triangular matrix as above and specifying the ranks of the left-justified submatrices in consecutive rows. (In the example above, the two strata depend on whether the submatrix in row $2$ and column $1$ has rank $1$ or $0$.) I don't know whether Allen has written up this fact.

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    $\begingroup$ Dear David, may I ask one more question? I wonder if there exist similar result for Schubert cell and opposite cell for general flag variety $G/P$? Where $P$ is a parabolic group. And for Deodhar decomposition in this case? @David Speyer $\endgroup$ Jun 13, 2022 at 15:52
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    $\begingroup$ Yes. If $u$ and $w$ are minimal coset representative in $W/W_P$, then $(B_+ w B_+ \cap B_- u B_+)/B_+$ projects isomorphically to $(B_+ w P \cap B_- u P)/P$. (I think I got all the indices in the right place :).) See mathoverflow.net/questions/5131 and, for the Grassmannian case, see arxiv.org/abs/1807.09229 . $\endgroup$ Jun 13, 2022 at 16:06
  • $\begingroup$ Dear David, I need a reference about the result"$(B_{+}wB_{+}\cap B_{−}uB_{+})/B_{+}$ projects isomorphically to $(B_{+}wP\cap B_{−}uP)/P$". I have seen the paper "Projections of Richardson Varieties" written by Knutson, Lam and you, but I haven't found such result in this paper. May you show me a reference about it, please? It seems important for me, since I need to cite it . Thank you very much! @David E Speyer $\endgroup$ Jun 16, 2022 at 7:07
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    $\begingroup$ Lemma 3.1 in our paper almost says this: It says that the map from the first space to G/P is injective, and the second space is defined as the image of that map. So Lemma 3.1 says that the natural map between these spaces is a bijection. The paragraph after Lemma 3.1 seems to assert more strongly that this is an isomorphism and cites two papers, by Rietsch and by Lusztig, so I would look in those papers. $\endgroup$ Jun 16, 2022 at 11:58
  • $\begingroup$ OK! I'll check it , thank you😊 $\endgroup$ Jun 16, 2022 at 15:23

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