Diagonal bases in Orlik-Solomon type algebras (Q1430519)

Let \({\mathcal A}\) be an arrangement of hyperplanes in \({\mathbb C}^\ell,\) and \(M={\mathbb C}^\ell - \bigcup_{H\in {\mathcal A}} H.\) \textit{A.~Szenes} [Int. Math. Res. Not. 1998, No. 18, 937--956 (1998; Zbl 0968.11015)] defined the iterated residue of a meromorphic function with poles along \(\bigcup_{H\in {\mathcal A}} H\), relative to an ordered set \(S\) of \(\ell\) linearly independent hyperplanes. A key technical tool is the notion of diagonal basis, which we will briefly describe here. An ordered independent set \(S=\{H_1,\ldots, H_k\}\subseteq {\mathcal A}\) determines a torus \((S^1)^k \subseteq M\), representing an element \(c_S \in H_k(M),\) and also a logarithmic \(k\)-form on \(M\), and hence an element \(\omega_S\) of \(H^k(M)\cong \Hom(H_k(M),{\mathbb C})\). The homology class \(c_S\) depends only on the flag \(H_1\supseteq H_1\cap H_2 \supseteq \cdots \supseteq \bigcap_{i=1}^k H_i.\) A collection \({\mathcal D}\) of independent sets of \(k\) hyperplanes forms a diagonal basis in degree \(k\) if the corresponding sets \(\{\omega_S \;| \;S \in {\mathcal D}\}\) and \(\{c_S \;| \;S \in {\mathcal D}\}\) form dual bases of \(H^k(M)\) and \(H_k(M)\) respectively. Equivalently, \(\langle \omega_S, c_{S'} \rangle = \delta_{S,S'}\) for all \(S,S' \in {\mathcal D}.\) Szenes proved that the collection of sets of size \(k\) containing no broken circuits (``\textbf{nbc} sets''), relative to any linear ordering of \({\mathcal A}\), form a diagonal basis for every \(k\). In the paper under review the authors give a formulation of the notions of iterated residue and diagonal basis for a general class of ``Orlik-Solomon type'' algebras, and prove that the \textbf{nbc} sets form a diagonal basis in this more general setting. Given a matroid \(\mathcal M\) on \(n\) points and a function \(\chi: 2^{[n]} \to {\mathbb K},\) the second author and \textit{M.~Las Vergnas} in [Eur. J. Comb. 22, No. 5, 699-704 (2001; Zbl 0984.52016)] defined a (commutative or skew-commutative) \({\mathbb K}\)-algebra \({\mathbb A}\) called a \(\chi\)-algebra, and proved that \textbf{nbc} sets yield a basis. This class of algebras includes as special cases the cohomology algebra \(H^*(M)\) above, isomorphic to the Orlik-Solomon algebra of the underlying matroid (\(\chi=1\)), the Orlik-Terao algebra of a set of linear forms (\(\chi = \det\)), and the oriented matroid analogue of the Orlik-Terao algebra introduced by the first author (\(\chi = \text{basis signature}\)). There is a deletion-contraction split short exact sequence for \(\chi\)-algebras. For Orlik-Solomon algebras the contraction homomorphism is a residue, so the authors define iterated residue in general as an iterated contraction. An ordered independent set \(S\) determines an element \(\omega_S \in {\mathbb A}\) and a flag \(c_S\) in the order complex of the lattice of flats of \(\mathcal M\). A collection \(\mathcal D\) of ordered independent sets is a diagonal basis in the general setting if the set of flags \(\{c_S \;| \;S \in {\mathcal D}\}\) satisfies a certain combinatorial condition which implies that \(\omega_S\) and the iterated residue corresponding to \(S\) are algebraic duals, for all \(S \in {\mathcal D}.\) It is shown that the \textbf{nbc} sets, relative to any linear ordering of the ground set, have this property. Uniform formulae for coordinates and iterated residues in terms of \textbf{nbc} bases are derived.