3
$\begingroup$

Let $x$ and $y$ be two elements of $S_n$. Let $U(x,y)$ be the intersection $(BxB \cap B_{-} y B)/B$ inside the flag variety. Here $B$ and $B_{-}$ are the groups of upper and lower-triangular matrices respectively. Kazhdan and Lusztig define the $R$ polynomial $R_{x,y}(q)$. As explained in Theorem 1.3 of Deodhar, one such description is that $$R_{x,y}(q) = \# U(x,y)(\mathbb{F}_q).$$

Now, $U(x,y)$ is an affine variety, so I don't get to know that the weight filtration matches the cohomological filtration. But, looking at examples, it looks like the coefficient of $(-1)^{\ell(x) - \ell(y) - i } q^i$ in $R_{x,y}(q)$ is the Betti number $\dim H^i(U(x,y))$. For example, with $n=2$, $x=(21)$ and $y$ being the identity, then $U(x,y)$ is the torus $\mathbb{A}^1 \setminus \{ 0 \}$, it has $q-1$ points, and its betti numbers are $\dim H^0 = \dim H^1 = 1$.

This is probably something very standard, but I didn't find it the first few places I tried, and it seemed faster to ask here than to keep looking. Thanks!

$\endgroup$
5
  • $\begingroup$ As you probably know, $\\#U(x,y)(\mathbb{F}_q)$ the supertrace of the Frobenius acting on $H^*_c(U(x,y))$; one possibility is that the eigenvalue of Frobenius on $H^{2\ell(x)-2\ell(y)-i}_c(U(x,y))$ is $q^{\ell(x)-\ell(y)-i}$. This would happen if the pullback to the torus in the Deodhar decomposition is injective in cohomology. $\endgroup$
    – Ben Webster
    Dec 19, 2011 at 17:51
  • $\begingroup$ Unfortunately, that pullback is not injective. I have an example in $GL_5$ with $R$-polynomial $q^6 - 4q^5 + 7q^4 - 8q^3 + 7q^2 - 4q +1$. If these are also betti numbers, then $\dim H^1(U)=4$ and $\dim H^2(U) = 7$. The image of $H^2(U)$ must land in the part of $H^2(T)$ generated by $H^1(U)$ (because pullback is a map of rings) and that part has dimension $\binom{4}{2} = 6$. $\endgroup$ Dec 19, 2011 at 18:03
  • $\begingroup$ I seem to recall seeing some paper by someone somewhere in Europe whose name I didn't recognize giving a geometric interpretation of R-polynomials, possibly in connection with Bott-Samelson resolutions or buildings or both. Unfortunately I don't remember enough to track it down. $\endgroup$ Dec 19, 2011 at 21:03
  • $\begingroup$ @Ben: Dear Ben, could you please let me knowthe fulln name of the Mathematician associated with $\text{Deodhar Decomposition}$. $\endgroup$
    – C.S.
    Dec 26, 2011 at 15:08
  • $\begingroup$ Vinay Vithal Deodhar, according to genealogy.math.ndsu.nodak.edu/id.php?id=9927 . The decomposition Ben is referring to is from ams.org/mathscinet-getitem?mr=782232 $\endgroup$ Dec 26, 2011 at 16:07

1 Answer 1

7
$\begingroup$

I think that this is one of these things that looks plausible in small examples but is false. For example, this would imply that the coefficients of R polynomials are alternating in $q$. This is implied by another conjecture called the Gabber-Joseph conjecture (roughly: coefficients of R-poynomials give dimensions of Ext groups between Verma modules), which is false. See "A counterexample to the Gabber-Joseph conjecture" by Brian Boe.

$\endgroup$
1
  • 3
    $\begingroup$ Besides correcting the spelling of Brian Boe's first name, I'll add a specific MathSciNet listing (it didn't get into arXiv): Boe, Brian D. (1-GA). A counterexample to the Gabber-Joseph conjecture. Kazhdan-Lusztig theory and related topics (Chicago, IL, 1989), 1–3, Contemp. Math., 139, Amer. Math. Soc., Providence, RI, 1992. I discussed this in Remark 8.11 of my 2008 book on the BGG category, hedging my bets at the end on whether there will be a nice interpretation of the R-polynomials. It still seems to be an open question. $\endgroup$ Dec 19, 2011 at 21:22

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.