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Poincaré polynomial of a class of signed complete graphic arrangements. (English) Zbl 1144.52025
Konno, Kazuhiro (ed.) et al., Algebraic geometry in East Asia—Hanoi 2005. Proceedings of the 2nd international conference on algebraic geometry in East Asia, Hanoi, Vietnam, October 10–14, 2005. Tokyo: Mathematical Society of Japan (ISBN 978-4-931469-45-7/hbk). Advanced Studies in Pure Mathematics 50, 289-297 (2008).
Following T. Zaslavsky [Discrete Appl. Math. 4, 47–74 (1982; Zbl 0476.05080)], the authors define a signed complete graph: A signed complete graph \(\Sigma_n = (K_n, \sigma)\) consists of an ordinary complete graph \(K_n\) with \(n\) vertices, and an arc labelling mapping \(\sigma: E\to \{\pm\}\), where \(E\) is the edge set of \(K_n\). Let \(E_+ = \sigma^{-1}(+)\) and \(E_{-} =\sigma^{-1}(-)\) denote the sets of the positive and negative edges respectively. An edge \(\{ij\}\in E_+\) is denoted by \(\{ij\}^+\) and is pictured as a line segment connecting the vertices \(i\) and \(j\). An edge \(\{ij\}\in E_-\) is denoted by \(\{ij\}^-\) and is pictured as a dashed line segment connecting the vertices \(i\) and \(j\).
Let \(V\) be an \(n\)-dimensional vector space over a field \(\mathbb K\). Let \(S\) be the symmetric algebra over the dual space \(V^* :=\operatorname{Hom}_{\mathbb K}(V, \mathbb K)\). If \(x_1, \dots, x_n\) is a basis of \(V^*\), then there are identifications \(S = \mathbb K[x_1,\dots, x_n]\) and \(V = \mathbb K^n\). A hyperplane \(H\) in \(\mathbb K^n\) is the zero set of a degree one polynomial in the variables \(x_1, \dots, x_n\). An arrangement \({\mathcal A}\) in \(\mathbb K^n\) is a finite collection of hyperplanes.
Given a signed complete graph \(\Sigma_n=(K_n, \sigma)\), define an arrangement \({\mathcal A}(\Sigma_n)\) in \(\mathbb K^n\) as follows:
\[ \{x_i-x_j = 0\} \in {\mathcal A}(\Sigma_n)\;\text{ if } \{ij\}\in E_+ \] \[ \{x_i+x_j=0\} \in {\mathcal A}(\Sigma_n)\;\text{ if } \{ij \} \in E_-. \] The authors compute the Poincaré polynomial of hyperplane arrangements associated with a class of signed complete graphs. They also make a factorization of the Poincaré polynomial over the integers.
For the entire collection see [Zbl 1135.14003].
52C35 Arrangements of points, flats, hyperplanes (aspects of discrete geometry)
06C05 Modular lattices, Desarguesian lattices
05C22 Signed and weighted graphs