2-roots for simply laced Weyl groups

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 $n \geq 6$),andaffine$E_6$,$E_7$and$E_8$\mathrm{.}Weshowthatanaturalcodimension-$1$submodule$M$ofthesymmetricsquareofthereflectionrepresentationof$W$hasaremarkablecanonicalbasis$$\mathcal{B}$$ 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.