Orthogonal roots, Macdonald representations, and quasiparabolic sets

TL;DR

This paper explores the structure of orthogonal roots in certain Weyl groups, revealing quasiparabolic sets and bases for Macdonald representations, and describes a special graph related to the E8 root system.

Contribution

It introduces the quasiparabolic set structure on maximal orthogonal positive roots and connects it to Macdonald representations and graph isomorphisms in E8.

Findings

Maximal orthogonal positive roots form a quasiparabolic set.

Nesting and crossing structures lead to new bases for Macdonald representations.

A graph related to E8 roots is described as quantum isomorphic to the orthogonality graph.

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.