Steven Klee

Lower Bound Theorems and a Generalized Lower Bound Conjecture for balanced simplicial complexes

A (d − 1)-dimensional simplicial complex is called balanced if its underlying graph admits a proper d-coloring. We show that many well-known face enumeration results have natural balanced analogs (or at least conjectural analogs). Specifically, we prove the balanced analog of the celebrated Lower Bound Theorem for normal pseudomanifolds and characterize the case of equality; we introduce and characterize the balanced analog of the Walkup class; we propose the balanced analog of the Generalized Lower Bound Conjecture and establish some related results. We close with constructions of balanced manifolds with few vertices.

Markov Chains for Promotion Operators

We consider generalizations of Schutzenberger’s promotion operator on the set \(\mathcal{L}\) of linear extensions of a finite poset. This gives rise to a strongly connected graph on \(\mathcal{L}\). In earlier work (Ayyer et al., J. Algebraic Combinatorics 39(4), 853–881 (2014)), we studied promotion-based Markov chains on these linear extensions which generalizes results on the Tsetlin library. We used the theory of \(\mathcal{R}\)-trivial monoids in an essential way to obtain explicitly the eigenvalues of the transition matrix in general when the poset is a rooted forest. We first survey these results and then present explicit bounds on the mixing time and conjecture eigenvalue formulas for more general posets. We also present a generalization of promotion to arbitrary subsets of the symmetric group.

Transportation Problems and Simplicial Polytopes That Are Not Weakly Vertex-Decomposable

Provan and Billera defined the notion of weak k-decomposability for pure simplicial complexes in the hopes of bounding the diameter of convex polytopes. They showed the diameter of a weakly k-decomposable simplicial complex Δ is bounded above by a polynomial function of the number of k-faces in Δ and its dimension. For weakly 0-decomposable complexes, this bound is linear in the number of vertices and the dimension. In this paper we exhibit the first examples of non-weakly 0-decomposable simplicial polytopes. Our examples are in fact polar to certain transportation polytopes.

Lower bounds for Buchsbaum* complexes

The class of (d-1)-dimensional Buchsbaum* simplicial complexes is studied. It is shown that the rank-selected subcomplexes of a (completely) balanced Buchsbaum* simplicial complex are also Buchsbaum*. Using this result, lower bounds on the h-numbers of balanced Buchsbaum* simplicial complexes are established. In addition, sharp lower bounds on the h-numbers of flag m-Buchsbaum* simplicial complexes are derived, and the case of equality is treated.

Face enumeration on simplicial complexes

In this chapter we survey many exciting developments on the face numbers of simplicial complexes from the past two decades. We focus on simplicial complexes whose geometric realizations are (homology) manifolds, as well as manifolds with additional combinatorial structure such as balanced manifolds or flag manifolds. The discussed results range from the Upper Bound Theorem for manifolds to the balanced Generalized Lower Bound Theorem for balanced polytopes.

Lower Bounds for Cubical Pseudomanifolds

It is verified that the number of vertices in a d-dimensional cubical pseudomanifold is at least 2d+1. Using Adin’s cubical h-vector, the generalized lower bound conjecture is established for all cubical 4-spheres, as well as for some special classes of cubical spheres in higher dimensions.

Obstructions to weak decomposability for simplicial polytopes

Author(s): Hahnle, Nicolai; Klee, Steven; Pilaud, Vincent | Abstract: Provan and Billera introduced notions of (weak) decomposability of simplicial complexes as a means of attempting to prove polynomial upper bounds on the diameter of the facet-ridge graph of a simplicial polytope. Recently, De Loera and Klee provided the first examples of simplicial polytopes that are not weakly vertex-decomposable. These polytopes are polar to certain simple transportation polytopes. In this paper, we refine their analysis to prove that these

A Lower Bound Theorem for Centrally Symmetric Simplicial Polytopes

d

FROM FLAG COMPLEXES TO BANNER COMPLEXES

-dimensional polytopes are not even weakly

Counting Binomial Coefficients Divisible by a Prime Power

O(\sqrt{d})