Tractable partially ordered sets derived from root systems and biased graphs
type: Item
instance of: scholarly article
title: Tractable partially ordered sets derived from root systems and biased graphs
zbMATH Open document ID: 0912.06003
DOI: 10.1023/A:1006086805922
author: Thomas Zaslavsky
author: Philip J. Hanlon
published in: Order
publication date: 1999-05-16T00:00:00Z
review text: 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 b…
reviewed by: J. Neggers
Mathematics Subject Classification ID: 06A07
Mathematics Subject Classification ID: 05C15
Mathematics Subject Classification ID: 05B35
Mathematics Subject Classification ID: 05C99
Mathematics Subject Classification ID: 52C35
zbMATH DE Number: 1210054
zbMATH Keywords: partially ordered set
zbMATH Keywords: graded poset
zbMATH Keywords: signed graph
zbMATH Keywords: gain graph
zbMATH Keywords: biased graph
zbMATH Keywords: chromatic polynomial
zbMATH Keywords: bias matroid
zbMATH Keywords: lift matroid
zbMATH Keywords: root system
zbMATH Keywords: arrangement of hyperplanes
zbMATH Keywords: Hilbert polynomial
zbMATH Keywords: Möbius function
zbMATH Keywords: characteristic polynomial
zbMATH Keywords: extrinsic grading function
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1210054