论文标题
2根简单的Weyl组
2-roots for simply laced Weyl groups
论文作者
论文摘要
我们介绍和研究“ 2根”，它们是Kac--Moody代数正交根的对称张量产物。我们专注于$ w $是一个简单的Y形dynkin图的Weyl组$ y_ {a，b，c} $具有$ n $ vertices，并设有三个任意有限长度$ a $ a $，$ b $和$ c $的分支；此的特殊情况包括类型$ d_n $，$ e_n $（用于任意$ n \ geq 6 $）和仿射$ e_6 $，$ e_7 $和$ e_8 $。我们表明，自然的codimension- $ 1 $ subpodule $ m $ $ w $的反射表示形式的对称平方具有出色的规范基础$ \ mathcal {b} $，由2 roots组成。我们证明，关于$ \ Mathcal {b} $，$ W $的每个元素都由列符号代数的列签名矩阵表示。如果$ W $是一个有限的Weyl组，则每个2根的$ W $ - 轨道具有最高的元素，类似于最高根，我们明确地计算了这些元素。我们证明，如果$ w $不具有仿射类型，则模块$ m $在特征零中完全可降低，并且其每个非平凡的直接汇总都由2架2个根的$ w $ orbit跨越。
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 $n \geq 6$), 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 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.
