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 $Y_{a,b,c}$ 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 conjecture that, with respect to $\mathcal{B}$, every element of $W$ is represented by a column sign-coherent matrix in the sense of cluster algebras, and we prove the conjecture in the finite and affine cases. We also   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.(This is joint work with Richard Green.)

会议密码:2022