Tractable partially ordered sets derived from root systems and biased graphs
Given a (finite) poset \(Q\) (with \(\widehat 0\)) and \(\widetilde h:Q\to N= \{0,1,2,\dots\}\) a strictly increasing function such that \(\widetilde h(\widehat 0)= 0\) and an assigned constant \(\widetilde h(Q)\geq \widetilde h(x)\) for all \(x\in Q\), the characteristic polynomial of \(\widetilde Q= (Q,\widetilde h,\widetilde h(Q))\) is defined to be \(p_{\widetilde Q}(\lambda)= \sum_{x\in Q}\mu_Q(\widehat 0,x) \lambda^{\widetilde h(Q)-\widetilde h(x)}\), where \(\mu_Q(\widehat 0, x)\) is the usual Möbius function of \(Q\). This function \(\widetilde h\) can represent many kinds of interesting information on \(Q\), such as height, rank, etc., with a corresponding characteristic polynomial. The extrinsic grading function \(\widetilde h\) is often taken to be the height function in relevant examples. Combinatorial information packed into such polynomials reflects deep properties of \(Q\) and may be analyzed in a variety of ways, as in ring-theoretic, algebraic-topological via the order-complex, via generating functions of various sorts other than the characteristic polynomial itself, functions related to diagrams associated with \(Q\) as a graph, and other points of view. Detailed information on \(p_{\widetilde Q}(\lambda)\) and the relation to the structure of \(Q\) is not normally very easy to come by even though interesting and revealing when so obtained. In this very substantial paper a class of posets is introduced and ``tamed'' using the full spectrum of paraphernalia alluded to in the previous paragraph. Given that \(B_n\) is the Boolean lattice of subsets of \([n]= \{1,\dots, n\}\), then the posets \(Q\) can be visualized as the lattice of flats spanned by subsets of \(B_n\) with certain of these removed. Using the computational techniques developed in a sequence of lemmas describing properties of the posets in a variety of cases, the authors, as recognized high-practitioners of their art, are able to demonstrate why a certain curious observed relationship between the exponents of root-systems \(D_n= \{\pm U_i\pm U_j, 1\leq i<j\leq n\}\) and \(D_n\cup\{\pm U_i, i\in[n]\}\) in \(\mathbb{R}^n\) obtains, which is the starting point of the entire investigation discussed in this paper. Among other byproducts are variant proofs and derivations of known results and applications to other areas related to the type of poset theory under discussion.
scientific article; zbMATH DE number 1210054