When studying the combinatorics and representation of Coxeter groups, I often find it irksome to compute KL- and R- polynomials from scratch on Maple or Sage. The time it consumes to generate a complete list for all pairs of elements of a Coxeter group soon becomes unbearable as the magnitude of the group increases.
Surprisingly, the only one I could find is a small one by Mark Goresky(link text) obtained in 1996. Is there a more comprehensive list of such polynomials?
It seems meaningful work to generate such lists, compared with calculating the next biggest known prime number.
Best Answer
I have placed data files containing the polynomials for $S_n$ with $n\in \{4,5,6,7,8,9\}$ here. The corresponding file for $S_{10}$ is a few gigabytes as a plain text file; I could certainly send that if you're interested.
This belongs as a response to Jim Humphrey's post, but I don't think I have the reputation for that, so: Patrick Polo's result (for which you can also find a combinatorial proof by Caselli ) is a wonderful, important result: Given a polynomial $f(q)$ of degree $d$ with constant term $1$ and nonnegative integer coefficients, Polo constructs a pair of permutations $x$ and $w$ living in some $S_N$ for which $P_{x,w}(q) = f(q)$. But $N = 1 + d + f(1)$. So even though $1+14q + 60q^2 + 96q^3 + 43q^4 + 4q^5$ appears as a Kazhdan-Lusztig polynomial for a pair of permutations in $S_{10}$, Polo's construction returns permutations in $S_{224}$. There's still a lot to be learned about what can happen for small $S_n$. Theory is going to be a crucial guide in studying these polynomials. But studying the data as one would do in the physical sciences is also, I think, going to play an important role.