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

  • Authors:

  • Murali K. Srinivasan

  • Affiliations:

  • Department of Mathematics, Indian Institute of Technology, Bombay, Powai, Mumbai 400076, India

  • Venue:
  • European Journal of Combinatorics
  • Year:
  • 2004

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Abstract

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 /2 is bijective (respectively, injective). We show that the q-analog of the n-cube is unitary semi-Peck.