A commutative algebra for oriented matroids.

*(English)*Zbl 1016.52014Let \({\mathcal A}\) be an arrangement of hyperplanes in \({\mathbb R}^\ell\), with linear defining forms \(\{\phi_1,\dots,\phi_n\}\). In his work on cohomology of local systems, K. Aomoto introduced the algebra \(AO({\mathcal A})\) of rational forms generated by \((\phi_{i_1}\cdots \phi_{i_k})^{-1}\) where \(\{i_1,\dots,i_k\}\) ranges over all independent subsets of \(\mathcal A\) [Sugaku Expo. 9, 99-116 (1996; Zbl 0787.33001)]. P. Orlik and H. Terao in [Nagoya Math. J. 134, 65-73 (1994; Zbl 0801.05019)] constructed a presentation of the commutative algebra \(AO({\mathcal A})\) similar to the more familiar (skew-commutative) Orlik-Solomon algebra \(OS({\mathcal A})\). A crucial difference is that the presentation of \(AO({\mathcal A})\) depends explicitly on the coefficients of dependence relations among the forms \(\phi_i\). By contrast, \(OS({\mathcal A})\) is defined in terms of the underlying matroid of \(\mathcal A\). Orlik and Terao ask whether in fact \(AO({\mathcal A})\) is determined by the underlying matroid. The main point of the paper under review is to show that this is not the case.

This result is accomplished by studying a closely related algebra \({\mathbb A}({\mathcal M})\), a combinatorial analogue of \(AO({\mathcal A})\) defined in terms of the oriented matroid \(\mathcal M\) associated with \(\mathcal A\). The algebras \({\mathbb A}({\mathcal M})\) and \(AO({\mathcal A})\) are isomorphic in case there is a spanning set of dependence relations among the \(\phi_i\) having all coefficients equal to \(\pm 1\) or \(0\). It is shown that the first examples of C. Eschenbrenner and M. Falk [J. Algebr. Comb. 10, 189-199 (1999; Zbl 0955.52010)], which have different underlying matroids, nevertheless have isomorphic \(\mathbb A\) algebras, as is the case also for their Orlik-Solomon algebras. These examples satisfy the condition stated above, and thus provide a negative answer to the question of Orlik and Terao.

The author also establishes a “no-broken-circuit” basis result for the oriented matroid algebra \({\mathbb A}({\mathcal M})\). An interesting notion of “combinatorial basis” is introduced: this is a basis \(\{b_1,\dots,b_n\}\) of \({\mathbb A}^1({\mathcal M})\) for which a product of distinct elements \(b_{i_1}\cdots b_{i_k}=0\) if and only if \(\{i_1,\dots,i_k\}\) is dependent in the underlying matroid. The same definition makes sense for \(AO({\mathcal A})\) and \(OS({\mathcal A})\). The algebra \({\mathbb A}({\mathcal M})\) is said to fix the underlying matroid if the only combinatorial bases are obtained by permutation of the standard basis. It is shown that, in case the algebra \({\mathbb A}({\mathcal M})\) fixes the underlying matroid, \({\mathbb A}({\mathcal M})\) actually determines the oriented matroid \({\mathcal M}\), and thus the homotopy type of the complexified complement \(M\).

This result is accomplished by studying a closely related algebra \({\mathbb A}({\mathcal M})\), a combinatorial analogue of \(AO({\mathcal A})\) defined in terms of the oriented matroid \(\mathcal M\) associated with \(\mathcal A\). The algebras \({\mathbb A}({\mathcal M})\) and \(AO({\mathcal A})\) are isomorphic in case there is a spanning set of dependence relations among the \(\phi_i\) having all coefficients equal to \(\pm 1\) or \(0\). It is shown that the first examples of C. Eschenbrenner and M. Falk [J. Algebr. Comb. 10, 189-199 (1999; Zbl 0955.52010)], which have different underlying matroids, nevertheless have isomorphic \(\mathbb A\) algebras, as is the case also for their Orlik-Solomon algebras. These examples satisfy the condition stated above, and thus provide a negative answer to the question of Orlik and Terao.

The author also establishes a “no-broken-circuit” basis result for the oriented matroid algebra \({\mathbb A}({\mathcal M})\). An interesting notion of “combinatorial basis” is introduced: this is a basis \(\{b_1,\dots,b_n\}\) of \({\mathbb A}^1({\mathcal M})\) for which a product of distinct elements \(b_{i_1}\cdots b_{i_k}=0\) if and only if \(\{i_1,\dots,i_k\}\) is dependent in the underlying matroid. The same definition makes sense for \(AO({\mathcal A})\) and \(OS({\mathcal A})\). The algebra \({\mathbb A}({\mathcal M})\) is said to fix the underlying matroid if the only combinatorial bases are obtained by permutation of the standard basis. It is shown that, in case the algebra \({\mathbb A}({\mathcal M})\) fixes the underlying matroid, \({\mathbb A}({\mathcal M})\) actually determines the oriented matroid \({\mathcal M}\), and thus the homotopy type of the complexified complement \(M\).

Reviewer: Michael J.Falk (Flagstaff)

##### MSC:

52C40 | Oriented matroids in discrete geometry |