On quotients of posets, with an application to the q-analog of the hypercube

Let G be a finite group having an order preserving and rank preserving action on a finite ranked poset P. Let P/G denote the quotient poset. A well known result in algebraic Sperner theory asserts that an order raising G-linear map on V (P) (the complex vector space with P as basis) satisfying the full rank property induces an order raising linear map on V (P/G), also satisfying the full rank property. In this paper we prove a kind of converse result that has applications to Boolean algebras and their cubical and q-analogs. For a finite ranked poset P, let L denote the Lefschetz order raising map taking an element to the sum of the elements covering it and let Pi , 0 ≤ i ≤ n, where n = rank(P), denote the set of elements of rank i . We say that P is unitary Peck (respectively, unitary semi-Peck) if the map Ln−2i : V (Pi ) → V (Pn−i ), i < n/2 is bijective (respectively, injective). We show that the q-analog of the n-cube is unitary semi-Peck. © 2003 Elsevier Ltd. All rights reserved. MSC (2000): Primary 05E25; Secondary 06A07, 06A11