# May Schubert cell intersection with opposite big cell polynomial count?

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

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.

• 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 Jun 13, 2022 at 15:52
• 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 . Jun 13, 2022 at 16:06
• 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 Jun 16, 2022 at 7:07
• 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. Jun 16, 2022 at 11:58
• OK! I'll check it , thank you😊 Jun 16, 2022 at 15:23