Some Aspects of Groups Acting on Finite Posets

Let P be a finite poset and G a group of automorphisms of P. The action of G on P can be used to define various linear representations of G, and we investigate how these representations are related to one another and to the structure of P. Several examples are analyzed in detail, viz., the symmetric group G, acting on a boolean algebra, CL,(q) acting on subspaces of an n-dimensional vector space over GF(q), the hyperoctahedral group B, acting on the lattice of faces of a cross-polytope, and G, acting on the lattice II, of partitions of an n-set. Several results of a general nature are also proved. These include a duality theorem related to Alexander duality, a special property of geometric lattices, the behavior of barycentric subdivision, and a method for showing that certain sequences are unimodal. In particular, we give what seems to be the simplest proof to date that the q-binomial coefftcient [ ‘:‘I has unimodal coefficients.