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Register a new userThin Posets, CW Posets, and Categorification
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Alex Chandler
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Motivated by generalizing Khovanovs categorification of the Jones polynomial, we study functors $F$ from thin posets $P$ to abelian categories $mathcal{A}$. Such functors $F$ produce cohomology theories $H^*(P,mathcal{A},F)$. We find that CW posets, that is, face posets of regular CW complexes, satisfy conditions making them particularly suitable for the construction of such cohomology theories. We consider a category of tuples $(P,mathcal{A},F,c)$, where $c$ is a certain ${1,-1}$-coloring of the cover relations in $P$, and show the cohomology arising from a tuple $(P,mathcal{A},F,c)$ is functorial, and independent of the coloring $c$ up to natural isomorphism. Such a construction provides a framework for the categorification of a variety of familiar topological/combinatorial invariants: anything expressible as a rank-alternating sum over a thin poset.
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We consider 3 (weighted) posets associated with a graph G - the poset P(G) of distinct induced unlabelled subgraphs, the lattice Omega(G) of distinct unlabelled graphs induced by connected partitions, and the poset Q(G) of distinct unlabelled edge-subgraphs. We study these posets given up to isomorphism, and their relation to the reconstruction conjectures. We show that when G is not a star or a disjoint union of edges, P(G) and Omega(G) can be constructed from each other. The result implies that trees are reconstructible from their abstract bond lattice. We present many results on the reconstruction questions about the chromatic symmetric function and the symmetric Tutte polynomial. In particular, we show that the symmetric Tutte polynomial of a tree can be constructed from its chromatic symmetric function. We classify graphs that are not reconstructible from their abstract edge-subgraph posets, and further show that the families presented here are the only graphs not Q-reconstructible if and only if the edge reconstruction conjecture is true. Let f be a bijection from the set of all unlabelled graphs to itself such that for all unlabelled graphs G and H, hom(G,H) = hom(f(G), f(H)). We conjecture that f is an identity map. We show that this conjecture is weaker than the edge reconstruction conjecture. Our conjecture is motivated by homomorphism cancellation results due to Lovasz.
For any graded poset $P$, we define a new graded poset, $mathcal E(P)$, whose elements are the edges in the Hasse diagram of P. For any group, $G$, acting on the boolean algebra, $B_n$, we conjecture that $mathcal E(B_n/G)$ is Peck. We prove that the conjecture holds for common cover transitive actions. We give some infinite families of common cover transitive actions and show that the common cover transitive actions are closed under direct and semidirect products.
The Pareto dominance relation of a preference profile is (the asymmetric part of) a partial order. For any integer n, the problem of the existence of an n-agent preference profile that generates the given Pareto dominance relation is to investigate the dimension of the partial order. We provide a characterization of a partial order having dimension n in general.
We consider the problem of determining the maximum order of an induced vertex-disjoint union of cliques in a graph. More specifically, given some family of graphs $mathcal{G}$ of equal order, we are interested in the parameter $a(mathcal{G}) = min_{G in mathcal{G}} max { |U| : U subseteq V, G[U] text{ is a vertex-disjoint union of cliques} }$. We determine the value of this parameter precisely when $mathcal{G}$ is the family of comparability graphs of $n$-element posets with acyclic cover graph. In particular, we show that $a(mathcal{G}) = (n+o(n))/log_2 (n)$ in this class.
Toric posets are cyclic analogues of finite posets. They can be viewed combinatorially as equivalence classes of acyclic orientations generated by converting sources into sinks, or geometrically as chambers of toric graphic hyperplane arrangements. In this paper we study toric intervals, morphisms, and order ideals, and we provide a connection to cyclic reducibility and conjugacy in Coxeter groups.
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