The determinant formula for a matroid bilinear form
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 the unsigned Möbius function and \(\beta(K)=(-1)^{r(K)}\sum_{K'\leq{}K}\mu(K')r(K')\) is the beta invariant. This paper is fairly self-contained, giving all the standard notions in matroid theory which are used in the discussion.
scientific article; zbMATH DE number 1042599