The determinant formula for a matroid bilinear form
type: Item
instance of: scholarly article
title: The determinant formula for a matroid bilinear form
zbMATH Open document ID: 0918.05038
author: O. Diekmann
published in: Advances in Mathematics
publication date: 1999-08-09T00:00:00Z
review text: In \textit{V. V. Schechtman} and \textit{A. N. Varchenko} [Invent. Math. 106, No. 1, 139-194 (1991; Zbl 0754.17024)], a bilinear form associated with a hyperplane arrangement was introduced and a formula for its determinant was derived, exhibiting its factorisation into linear factors. The interest in such bilinear forms stems from the relation between the topology and differential geometry of the Knizhnik-Zamolodchikov equation arising in conformal field theory. In that relation, there is a natural connection with representation theory of Lie algebras and indeed the bilinear forms may be considered as analogues of the Shapovalov form for a representation with highest weight [see \textit{V. G. Kac}, Infinite-dimensional Lie algebras. 3rd ed. CUP (1990; Zbl 0716.17022)]. Supose \(M\) is a simple matroid of rank \(r\). Its flag space \(\mathcal F\) is defined to be the quotient of the free \(\mathbb{Z}\)-module on the set of all (not necessarily complete) flags in \(M\), by relations given for each flag with an ``\(i\)''-gap, \(\sum F=0\) where the sum is over all flags which contain the fixed flag obtained by adding an arbitrary flat at level \(i\). A \(k\)-tuple of points \(A=(p_1,\ldots,p_k)\) is said to be adjacent to a \(k\)-flag \(F=[\emptyset<K_1<\cdots<K_k]\) iff \(K_i=p_1\vee\cdots\vee{}p_i\) for all \(i\), written \(A\sim{}F\). The bilinear form of the matroid is a map \(B:{\mathcal F}\otimes{\mathcal F}\rightarrow{}R\) dependent on paramenters \(a_i=a(p_i)\) in a ring \(R\) over the integers, one given for each point \(p_i\) in \(M\), defined by \(B(F,F')=\sum\text{sgn}(\sigma)a_{i_1}\cdots{}a_{i_k}\), where the sum is over all \(i_j\)'s and permutations \(\sigma\) for which \((p_{i_1},\ldots,p_{i_k})\sim{}F\) and \((p_{\sigma(i_1)},\ldots,p_{\sigma(i_k)})\sim{}F'\). The main result of the paper is that \(\det B=\prod_Ka(K)^{\beta(K)\mu^+(M/K)}\), where the product is over all flats \(K\) of non-zero rank \(r(K)\) while \(a(K)=\prod{}a_{i_j}\), \(\mu^+\) is…
Mathematics Subject Classification ID: 05B35
Mathematics Subject Classification ID: 33C80
Mathematics Subject Classification ID: 52C35
zbMATH DE Number: 1042599
zbMATH Keywords: matroid
zbMATH Keywords: flag
zbMATH Keywords: bilinear form
reviewed by: Ruth Lawrence
MaRDI profile type: Publication
full work available at URL: aima.1996.1530
OpenAlex ID: W1970512471
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DOI: 10.1006/AIMA.1996.1530
description: scientific article; zbMATH DE number 1042599