2-roots for simply laced Weyl groups

TL;DR

This paper introduces 2-roots, a new concept related to symmetrized tensor products of roots in simply laced Weyl groups, revealing a canonical basis and sign-coherent matrix representations, with explicit calculations for finite types.

Contribution

It defines 2-roots and constructs a canonical basis for a submodule of the symmetric square of the reflection representation, connecting to cluster algebra structures and explicitly analyzing orbit elements.

Findings

Existence of a canonical basis of 2-roots in a codimension-1 submodule.

Representation of Weyl group elements by sign-coherent matrices.

Complete reducibility of the module in characteristic zero for non-affine types.

Abstract

We introduce and study "2-roots", which are symmetrized tensor products of orthogonal roots of Kac--Moody algebras. We concentrate on the case where $W$ is the Weyl group of a simply laced Y-shaped Dynkin diagram $Y_{a,b,c}$ having $n$ vertices and with three branches of arbitrary finite lengths $a$, $b$ and $c$; special cases of this include types $D_n$, $E_n$ (for arbitrary $n \geq 6$), and affine $E_6$, $E_7$ and $E_8$. We show that a natural codimension-$1$ submodule $M$ of the symmetric square of the reflection representation of $W$ has a remarkable canonical basis $\mathcal{B}$ that consists of 2-roots. We prove that, with respect to $\mathcal{B}$, every element of $W$ is represented by a column sign-coherent matrix in the sense of cluster algebras. If $W$ is a finite simply laced Weyl group, each $W$-orbit of 2-roots has a highest element, analogous to the highest root, and we calculate these elements explicitly. We prove that if $W$ is not of affine type, the module $M$ is completely reducible in characteristic zero and each of its nontrivial direct summands is spanned by a $W$-orbit of 2-roots.