Tianyuan Xu (Haverford College)

Algebra & Geometry Seminar

Monday, September 25th, 2023

Time: 4:30 p.m.  Place: Jeffery Hall, Room 422

Speaker: Tianyuan Xu (Haverford College)

Title: 2-roots for simply-laced Weyl groups.

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 E_6 , E_7 and E_8.

We construct a natural codimension-1 submodule M of the symmetric square of the reflection representation of W, as well as a canonical basis B of M that consists of 2-roots. The 2-roots of W and the basis B share many similarities with the roots and the simple roots of W. In particular, we show that every 2-root is a linear combination of the basis B with coefficients of like sign. We also describe a natural partial order on the set of 2-roots and explicitly describe the "highest 2-roots", the maximal 2-roots with respect to this order. (This is joint work with Richard Green.)