Quotients of Peck Posets

Abstract. An elementary, self-contained proof of a result of Pouzet and Rosenberg and of Harper is given. This result states that the quotient of certain posets (called unitary Peck) by a finite group of automorphisms retains some nice properties, including the Sperner property. Examples of unitary Peck posets are given, and the techniques developed here are used to prove a result of Lovhz on the edge-reconstruction conjecture. AMS (MOS) subject classifications (1980). Primary 06AlO; secondary.05C60, 20B25. Key words. Poset, Peck poset, Sperner property, automorphism group, quotient poset, edge-reconstruc-tion. Let P be a finite graded poset of rank n, i.e., P is a disjoint union of subsets PO, Pl, *a*, P,,, called ranks, such that if x E Pi and y covers x, then y E Pi+ 1. Let pi = /PiI, where P is said to be rank-symmetric if Pi =pn-i for all i, and rank-unimodal if PO < p1 < ** ’ <pi 2 Pj+l>pj+s 2 +.-> pn for some j. P satisfies the Sperner property if no antichain ( = set of pairwise incomparable elements) of P is bigger than the largest rank. More generally, P is k-Sperner if no union of k antichains is larger than the union of the k largest ranks, and is strongZy Sperner if it is kSperner for 1 < k < n t 1. P is a Peck poset if it is rank-symmetric, rank-unimodal, and strongly Sperner. Let Vi be the complex vector space with basis Pi. It is known [ 131, Lemma 1.1, that a finite graded poset P of rank n is Peck if and only if there exist linear transformations $9: vi-+ vi+1, 0 < i < n, satisfying: (A) IfxEPi then GiCx) =,E$+, ‘Yey y>‘