Orthogonal roots, Macdonald representations, and quasiparabolic sets

2024/09/03 by R. M. Green, Green, R. M., Tianyuan Xu +1 · 2 citations
Computer Science · Mathematics · #05E10 #06A07 #Advanced Combinatorial Mathematics #Combinatorics (math.CO) #FOS: Mathematics #Group Theory (math.GR) #Primary: 20F55 #Representation Theory (math.RT) #Secondary: 05E18 #Topological and Geometric Data Analysis

paper · pdf · doi:10.48550/arxiv.2409.01948

openalex publication_date 2024/09/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let W be a simply laced Weyl group of finite type and rank n. If W has type E₇, E₈, or D_n for n even, then the root system of W has subsystems of type nA₁. 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 E₈ to give a concise description of a graph that is known to be non-isomorphic but quantum isomorphic to the orthogonality graph of the E₈ root system.

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