Séminaire Lotharingien de Combinatoire, 93B.123 (2025), 12 pp.
Richard M. Green and
Tianyuan Xu
Orthogonal Roots, Macdonald Representations, and Quasiparabolic W-Sets
Abstract.
Let W be a finite Weyl group with root system Φ and of rank n>1.
We study the maximal sets of orthogonal positive roots of Φ with
cardinality n, which exist if and only if W has type E7, E8, or Dn for
n even. We show that in these types, the set X of all such maximal
orthogonal sets forms a quasiparabolic W-set in the sense of Rains-Vazirani.
The quasiparabolic structure can be described in terms of certain quadruples of
orthogonal roots that we call crossings, nestings, and alignments. This leads to
noncrossing and nonnesting bases of a suitable irreducible representation of W
known as a Macdonald representation, as well as some highly structured partially
ordered sets, including the strong Bruhat poset of symmetric groups. In type
E8, we use the set X 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.
Received: November 15, 2024.
Accepted: February 15, 2025.
Final version: April 1, 2025.
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