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A Note on the Orlik–Solomon Algebra

✍ Scribed by Raul Cordovil; Gwihen Etienne

Publisher
Elsevier Science
Year
2001
Tongue
English
Weight
112 KB
Volume
22
Category
Article
ISSN
0195-6698

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✦ Synopsis

Let M = M(E) be a matroid on a linear ordered set E. The Orlik-Solomon Z-algebra OS(M) of M is the free exterior Z-algebra on E, modulo the ideal generated by the circuit boundaries. The Z-module OS(M) has a canonical basis called 'no broken circuit basis' and denoted nbc. Let e X = e i , e i ∈ X ⊂ E. We prove that when e X is expressed in the nbc basis, then all the coefficients are 0 or ±1. We present here an algorithm for computing these coefficients. We prove in appendix a numerical identity involving the dimensions of the algebras of Orlik-Solomon of the minors of a matroid and its dual.

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