2-roots for simply laced Weyl groups 03/20/2023

Speaker: Tianyuan Xu, Haverford
Date and time: Monday, March 20, 4–5 pm
Place: Rome 206

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 with three branches of arbitrary finite lengths a, b and c; special cases of this include types D_n, E_n (for arbitrary n ≥ 6), and affine E₆, E₇ and E₈.

With motivations from Kazhdan–Lusztig theory, we construct a natural codimension-1 submodule M of the symmetric square of the reflection representation of W, as well as a canonical basis $\mathcal{B}$ of M 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. We also prove that if W is not of affine type, then 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. Finally, we define and describe highest 2-roots, which are analogous to highest roots of usual root systems. (This is joint work with Richard Green.)