We generalize results of Calderbank, Hanlon and Robinson on the representation of the symmetric group on the homology of posets of partitions with restricted block size. Calderbank, Hanlon and Robinson consider the cases of block sizes that are congruent to 0 mod d and 1 mod d for fixed d . We derive a general formula for the representation of the symmetric group on the homology of posets of pa...

We explore the connection between polygon posets, which is a class of ranked posets with an edgelabeling which satisfies certain 'polygon properties', and the weak order of Coxeter groups. We show that every polygon poset is isomorphic to a join ideal in the weak order, and for Coxeter groups where no pair of generators have infinite order the converse is also true. The class of polygon posets ...

In this paper we present a bijection between composition matrices and (2+ 2)free posets. This bijection maps partition matrices to factorial posets, and induces a bijection from upper triangular matrices with non-negative entries having no rows or columns of zeros to unlabeled (2+ 2)-free posets. Chains in a (2+ 2)-free poset are shown to correspond to entries in the associated composition matr...

The involutory dimension, if it exists, of an involution poset P := (P,≤,′ ) is the minimum cardinality of a family of linear extensions of ≤, involutory with respect to ′, whose intersection is the ordering ≤. We show that the involutory dimension of an involution poset exists iff any pair of isotropic elements are orthogonal. Some characterizations of the involutory dimension of such posets a...

It has been an open problem to characterize posets P with the property that every order-preserving map on P has a fixed point. We give a characterization of such posets in terms of their retracts.

We consider Markov chains on partially ordered sets that generalize the success-runs and remaining life chains in reliability theory. We find conditions for recurrence and transience and give simple expressions for the invariant distributions. We study a number of special cases, including rooted trees, uniform posets, and posets associated with positive semigroups. 1 Partially Ordered Sets 1.1 ...

For von Neumann algebras M,N not isomorphic to C C and without type I2 summands, we show that for an order-isomorphism f : AbSub M! AbSub N between the posets of abelian von Neumann subalgebras of M and N , there is a unique Jordan ⇤-isomorphism g : M! N with the image g[S] equal to f(S) for each abelian von Neumann subalgebra S of M. The converse also holds. This shows the Jordan structure of ...

A class of ranked posets {(D h, )} has been recently defined in order to analyse, from a combinatorial viewpoint, particular systems of real homogeneous inequalities between monomials. In the present paper we focus on the posets D 2 , which are related to systems of the form {xaxb ∗abcd xcxd : 0 ≤ a, b, c, d ≤ k, ∗abcd ∈ {<, >}, 0 < x0 < x1 < ... < xk}. As a consequence of the general theory, t...