Shellability is a well-known combinatorial criterion on a simplicial complex ∆ for verifying that the associated Stanley-Reisner ring k[∆] is Cohen-Macaulay. A notion familiar to commutative algebraists, but which has not received as much attention from combinatorialists as the Cohen-Macaulay property, is the notion of a Golod ring. Recently, Jöllenbeck introduced a criterion on simplicial comp...

We present a class of subposets of the partition lattice 5n with the following property: The order complex is homotopy equivalent to the order complex of 5n−1, and the Sn-module structure of the homology coincides with a recently discovered lifting of the Sn−1-action on the homology of5n−1. This is the Whitehouse representation on Robinson’s space of fully-grown trees, and has also appeared in ...

Given a finite group G and a natural number n, we study the structure of the complex of nested sets of the associated Dowling lattice Qn(G) (Proc. Internat. Sympos., 1971, pp. 101–115) and of its subposet of the G-symmetric partitions Qn(G) which was recently introduced by Hultman (http://www.math.kth.se/ ~hultman/, 2006), together with the complex of G-symmetric phylogenetic trees T G n . Hult...

Generalizing the concepts of Stanley–Reisner and affine monoid algebras, one can associate to a rational pointed fan Σ in Rd the Zd-graded toric face ring K[Σ]. Assuming that K[Σ] is Cohen–Macaulay, the main result of this paper is to characterize the situation when its canonical module is isomorphic to a Zd-graded ideal of K[Σ]. From this result several algebraic and combinatorial consequences...

The torus manifolds have been defined and studied by Masuda and Panov ([7]) who in particular also describe its cohomology ring structure. In this note we shall describe the topological K-ring of a class of torus manifolds (those for which the orbit space under the action of the compact torus is a homology poytope whose nerve is shellable) in terms of generators and relations. Since these torus...

The purpose of this paper is to extend to monoids the work of Björner, Wachs and Proctor on the shellability of the Bruhat-Chevalley order on Weyl groups. Let M be a reductive monoid with unit group G, Borel subgroup B and Weyl group W . We study the partially ordered set of B×Borbits (with respect to Zariski closure inclusion) within a G × G-orbit of M . This is the same as studying a W ×W -or...

The absolute order on the hyperoctahedral group Bn is investigated. Using a poset fiber theorem, it is proved that the order ideal of this poset generated by the Coxeter elements is homotopy Cohen–Macaulay. This method results in a new proof of Cohen–Macaulayness of the absolute order on the symmetric group. Moreover, it is shown that every closed interval in the absolute order on Bn is shellab...

A vector algorithm is an algorithm that applies a bounded number of vector operations to an input vector, regardless of the length of the input. The allowable operations are usually restricted to bit-wise operations available in processors, including shifts and binary addition with carry. This paper studies automata for which it is easy to derive a vector algorithm to compute the sequence of st...