Reduced decomposition for Weyl group elements which support a Bessel function

Question

Let $\Delta$ be a set of simple roots for a reduced root system, and let $(W,S)$ be the associated Coxeter system, where $W$ is the Weyl group and $S$ is the set of simple reflections corresponding to the elements of $\Delta$.

For each $\theta \subseteq \Delta$, there is a unique element $w_0 \in W$ such that $w_0(\theta) \subseteq \Delta$, and $w_0(\Delta - \theta) < 0$. Explicitly, $w_0$ is equal to $w_{\ell} w_{\ell}^{\theta}$, where $w_{\ell}$ is the long element of $W$, and $w_{\ell}^{\theta}$ is the long element of the root system corresponding to $\theta$.

An element of $W$ is said to support a Bessel function if it is of the form $w_{\ell} w_{\ell}^{\theta}$ for some $\theta \subseteq \Delta$.

I'm interested in finding reduced decompositions for Weyl group elements which support Bessel functions, in particular for the root system $C_n$. Is there a database or table of such reduced decompositions? I know there are tables of reduced decompositions of long elements of root systems, but I have never seen tables for other elements.

Answer 1

I'd be extremely surprised if such tables or database existed, mainly because the number of possible reduced decompositions for a Weyl group element tenda to grow very large as the rank increases. Even for the well-studied symmetric groups, this poses a serious problem. (It also makes tables of Kazhdan-Lusztig polynomials tricky to assemble; in Mark Goresky's old tables, he finds it convenient to work with a particular choice of reduced expression even though this makes it very complicated to compare other choices.)

In your case the Weyl group elements are more specialized but also tend to have huge numbers of reduced expressions as the rank grows. Ditto for the elements of $W$ which support Bessel functions.

A smaller comment: It's legal to invent your own notation, but it's much easier to follow Bourbaki (or another source if necessary). The Bourbaki notation is $w_0$ for the longest element of $W$ while $w_\theta$ is often used for the longest element of the parabolic subgroup $W_\theta$ relative to the chosen simple roots. (Also, a couple of your sentences need to be rewritten, though one can guess what you meant to say by "reduced decompositions of long elements of root systems".)