2-roots for simply laced Weyl groups

2022/04/20 by R. M. Green, Green, R. M., Tianyuan Xu +1 · 1 citation
Mathematics · #17B22 #20F55 #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #Algebraic structures and combinatorial models #FOS: Mathematics #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.2204.09765

openalex publication_date 2022/04/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 ≥ 6), and affine E₆, E₇ and E₈. We show that a natural codimension-1 submodule M of the symmetric square of the reflection representation of W has a remarkable canonical basis B that consists of 2-roots. We prove that, with respect to 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.

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• Orthogonal roots, quantum Hafnians, and generalized Rothe diagrams

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