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A commutative algebra for oriented matroids. (English) Zbl 1016.52014
Let $${\mathcal A}$$ 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 $$M$$.

##### MSC:
 52C40 Oriented matroids in discrete geometry
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