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## Lattice paths and the quadratic coefficient of Kazhdan-Lusztig polynomials

SPUR is finally over! Instead of continuing my series of blog posts, I thought I’d just link to my paper, Lattice paths and the quadratic coefficient of Kazhdan-Lusztig polynomials, and the first few blog posts should more or less provide enough background to read it.

My project ended up changing direction. The formula I was working with for the quadratic coefficient was so unwieldy that I ended up spending the whole time trying to simplify it, and instead of saying anything about non-negativity I ended up saying something about combinatorial invariance. The combinatorial invariance conjecture, which goes back to Lusztig and, independently, Dyer, says that the Kazhdan-Lusztig polynomial $P_{u,v}(q)$ depends only on the poset structure of $[u, v]$. In the special case that $u = e$ this was proven in 2006 by Brenti, Caselli, and Marietti. However, the conjecture is still open in general.

In particular, explicit nonrecursive formulas in which each term only depends on poset-theoretic data are not known in general. They are known in the case that the length $\ell(u, v)$ of the interval $[u, v]$ is less than or equal to $4$, and there is also such a formula for the coefficient of $q$ of $P_{e,v}(q)$ where $e$ is the identity. The main result of the paper is a formula for the coefficient of $q^2$ of $P_{u,v}(q)$ in which all but three of the terms depend only on poset data, which is a simplification of a general formula due to Brenti for $P_{u,v}(q)$ in terms of lattice paths. It reduces to

• a formula for the coefficient of $q^2$ of $P_{e,v}(q)$ in which all but one of the terms depends only on poset data,
• a formula for the coefficient of $q^2$ of $P_{u,v}(q)$ where $\ell(u, v) = 5$ in which all but one of the terms (but a different term) depends only on poset data (not in the paper), and
• a formula for the coefficient of $q^2$ of $P_{e,v}(q)$ where $\ell(u, v) = 6$ in which every term depends only on poset data.

I believe these formulas are known in some form, but the method of proof is likely to be novel. In any case, the troublesome terms in the above are all essentially coefficients of R-polynomials. If I revisit this project in the future, I will be focusing my attention on these coefficients, and my goal will be to find a poset-theoretic formula for $P_{u,v}(q)$ in the length $5$ case, the smallest-length case where (to my knowledge) combinatorial invariance is open.

### 2 Responses

1. Dear Qiaochu, Indeed, both the nonnegativity and the combinatorial invariance are extremely interesting questions about K-L and R- polynomials. A related problem is to try to extend the definition of K-L and R-polynomials to larger classes of regular CW-spheres (perhaps even some non regular CW spheres). When the CW sphere is a lattice (in the order theoretic sense) this is ok (except that nonnegativity is still a huge problem). I am somehow pessimistic that one can extend K-L polynomials to all regular CW spheres but maybe one can identify interesting classes which are larger than those with the lattice property so that the R-polynomials are non trivial.

Anyway, it is great that you are blogging on these questions.

(I came accross in lectures more general polynomials called K-L-Vogan polynomials but I do not know what they really are; also curiously various far reaching extentions of K-L polynomials enters into Mulmuley’s geometric approach to lower bounds in complexity but, again, I am not sure what these extensions are…)

2. Dear Gil and Qiaochu,
Brenti-Caselli-Marietti and Marietti extended the definition of K-L and R-polynomials to a larger class and in these more general settings both the nonnegativity and the combinatorial invariance fail.
I hope this will be useful.
Luca

http://imrn.oxfordjournals.org/content/2006/29407