# Reduced decomposition for Weyl group elements which support a Bessel function

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.

• Where does this Bessel terminology come from? – LSpice Feb 7 '18 at 2:48

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".)