Decomposition of the longest element of the Weyl group using factors corresponding to the highest roots

TL;DR

This paper presents a novel decomposition of the longest element in finite Weyl groups into orthogonal root reflections, linking it to highest roots and the cascade of orthogonal roots, with explicit listings for each root system type.

Contribution

It introduces a new decomposition method for the longest element in Weyl groups using highest roots and establishes its uniqueness and connections to existing root cascade concepts.

Findings

Decomposition into orthogonal root reflections is possible for all root system types.

The decomposition is unique for each root system.

Subsets of highest roots align with the cascade of orthogonal roots used in algebra calculations.

Abstract

Let $\varPhi$ be a root system of a finite Weyl group $W$ with simple roots $\Delta$ and corresponding simple reflections $S$. For $J \subseteq S$, denote by $W_J$ the standard parabolic subgroup of $W$ generated by $J$, and by $\Delta_J \subseteq \Delta$ the subset corresponding to $J$. We show that the longest element of $W$ is decomposed into a product of several ($\le |\Delta|$) reflections corresponding to mutually orthogonal roots, each of which is either the highest root of some subset $\Delta_J \subseteq \Delta$ or is a simple root. For each type of the root system, the factors of the specified decomposition are listed. The relationship between the longest elements of different types is found out. The uniqueness of the considered decomposition is shown. It turns out that subsets of highest roots, which give the decomposition of longest elements in the Weyl group, coincide with the cascade of orthogonal roots constructed by B.Kostant and A.Joseph for calculations in the universal enveloping algebra.