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 \(n \ge 6\)), 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 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.
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Acknowledgements
We thank Robert B. Howlett for suggesting to us a proof of Theorem 5.2 on which our current proof is based, and we thank the referee for reading the paper carefully and suggesting many improvements. We also thank Justin Willson for some helpful conversations
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Green, R.M., Xu, T. 2-roots for Simply Laced Weyl Groups. Transformation Groups (2023). https://doi.org/10.1007/s00031-023-09809-0
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DOI: https://doi.org/10.1007/s00031-023-09809-0