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. 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 mapLn−2i:V(Pi)→V(Pn−i), i<n/2is bijective (respectively, injective). We show that the q-analog of the n-cube is unitary semi-Peck.