Fixed Points of Zircon Automorphisms

A zircon is a poset in which every principal order ideal is finite and equipped with a so-called special matching. We prove that the subposet induced by the fixed points of any automorphism of a zircon is itself a zircon. This provides a natural context in which to view recent results on Bruhat orders on twisted involutions in Coxeter groups. 1. Background and results Let P be a partially ordered set (poset). A matching on P is an involution M : P → P such that M(p) ⊳ p or p ⊳ M(p) for all p ∈ P , where ⊳ denotes the covering relation of P . In other words, M is a graph-theoretic (complete) matching on the Hasse diagram of P . Definition 1.1. Suppose M is a matching on a poset P . Then, M is called special if for all p, q ∈ P with p⊳ q, we either have M(p) = q or M(p) < M(q). The term “special matching” was coined by Brenti [3]. In the context of an Eulerian poset, a special matching is another way to think of a compression labelling as defined by du Cloux [6]. Definition 1.2. A poset P is a zircon if for any non-minimal element x ∈ P , the subposet induced by the principal order ideal {p ∈ P | p ≤ x} is finite and has a special matching. Zircons were defined by Marietti in [12]. Actually, his definition differs somewhat from ours, but Proposition 2.3 below shows that they are equivalent. We have chosen to use our definition because it is typically more convenient to check the finiteness condition in Definition 1.2 rather than finding a rank function as required by the definition in [12]. The motivation to introduce zircons comes from the fact that they mimic the behaviour of Coxeter groups ordered by the Bruhat order. More precisely, the Bruhat order ideal below a non-identity element w in a Coxeter group has a special matching given by multiplication with any descent of w. The finiteness condition in Definition 1.2 is trivially satisfied, implying that the Bruhat order on any Coxeter group is a zircon. In this note, we study the fixed points of automorphisms of zircons. Our main results are the next theorem and its corollary. The proofs are postponed to Section 2. Say that a poset is bounded if it has unique maximal and minimal elements. Theorem 1.3. Suppose P is a finite, bounded poset equipped with a special matching M . Let φ be an automorphism of P . Then, the subposet of P induced by the fixed points of φ has a special matching.