Title: Orthogonal roots, Macdonald representations, and quasiparabolic sets Show Chinese title

Title: 正交根，麦克唐纳表示，准抛物集

Abstract: Let $W$ be a simply laced Weyl group of finite type and rank $n$. If $W$ has type $E_7$, $E_8$, or $D_n$ for $n$ even, then the root system of $W$ has subsystems of type $nA_1$. This gives rise to an irreducible Macdonald representation of $W$ spanned by $n$-roots, which are products of $n$ orthogonal roots in the symmetric algebra of the reflection representation. We prove that in these cases, the set of all maximal sets of orthogonal positive roots has the structure of a quasiparabolic set in the sense of Rains--Vazirani. The quasiparabolic structure can be described in terms of certain quadruples of orthogonal positive roots which we call crossings, nestings, and alignments. This leads to nonnesting and noncrossing bases for the Macdonald representation, as well as some highly structured partially ordered sets. We use the $8$-roots in type $E_8$ to give a concise description of a graph that is known to be non-isomorphic but quantum isomorphic to the orthogonality graph of the $E_8$ root system. Show Chinese abstract

Abstract: 设 $W$ 是一个有限型和秩为 $n$ 的简单反射 Weyl 群。 如果当$n$为偶数时，$W$的类型为$E_7$、$E_8$或$D_n$，则$W$的根系有类型为$nA_1$的子系统。 这给出了由$n$-根（即反射表示的对称代数中$n$正交根的乘积）张成的$W$的不可约 Macdonald 表示。 我们证明，在这些情况下，所有极大正交根集的集合具有 Rains--Vazirani 意义下的拟抛物结构。 这种拟抛物结构可以用我们称为横跨、嵌套和排列的特定正交根四元组来描述。 这导致了 Macdonald 表示的非嵌套和非横跨基，以及一些高度结构化的偏序集。 我们利用类型$E_8$中的$8$-根，简洁地描述了一个已知与$E_8$根系的正交图量子同构但非同构的图。