3
$\begingroup$

Let $[u, v]$ be a Bruhat interval in some Coxeter group. Let $I$ be the set of all Bruhat intervals. I am interested in functions $I \to \mathbb{Z}$ which are invariant under poset isomorphisms. Some obvious functions of this type include the functions $v_k$, the number of elements of rank $k$ (where the bottom element has rank $0$), and $e_k$, the number of edges between elements of rank $k$ and elements of rank $k+1$. Similarly, the function $n$, the length of the interval, allows us to write down the functions $v_{n-k}$ and $e_{n-k}$ for some fixed $k$.

Question: Are there any linear relationships between these functions other than the "obvious" ones?

The obvious ones include that $e_0 = v_1, e_{n-1} = v_{n-1}$, and $v_0 = v_n = 1$. Somewhat less obviously, the classification of Bruhat intervals of length $2$ implies that $e_1 = 2v_2, e_{n-2} = 2v_{n-2}$. Beyond that, I more or less expect that these functions are linearly independent - in particular I expect that the functions $v_i$ and $v_{n-i}$ are linearly independent. Are there good reasons to expect otherwise?

(Motivation: linear combinations of these functions occur in some formulas for small coefficients of Kazhdan-Lusztig polynomials that I'm working with, and I want to be sure my intuition that I can't simplify them any further is justified.)

Edit: There is one more relation $v_0 - v_1 + v_2 \mp ... = 0$ coming from the fact that Bruhat intervals are Eulerian. But I should say that I'm only interested in the case where $i$ is small compared to $n$, so this relation doesn't affect what I'm looking at.

$\endgroup$
  • $\begingroup$ Should we say that the functions you are interested in are all parts of the flag f-vector? Can we rephrase your question as: what are defining equations of the subspace of flag f-vectors spanned by Bruhat intervals? Maybe we can then pass to the right family polytopes to think about this? $\endgroup$ – Alexander Woo Jul 21 '10 at 20:22
  • 4
    $\begingroup$ Since Bruhat intervals are Eulerian their flag $f$-vectors satisfy the Bayer-Billera relations. These give many relations of the type you are looking for. For instance, Gil Kalai gives a few examples at gilkalai.wordpress.com/2008/06/22/…. I doubt whether the flag $f$-vectors of Bruhat intervals satisfy any linear relations not already a consequence of the Eulerian property. $\endgroup$ – Richard Stanley Jul 21 '10 at 21:38
  • $\begingroup$ Thanks for the link! In the notation of that post, I am only looking for relations between the f_i and the f_{i,i+1}, and it doesn't look like the argument described gives many such relations. But I'll have to take a closer look at the Bayer-Billera relations. $\endgroup$ – Qiaochu Yuan Jul 21 '10 at 22:30

Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Browse other questions tagged or ask your own question.