The Varchenko-Gel’fand ring of a cone. (English) Zbl 1504.52021
Summary: For a hyperplane arrangement in a real vector space, the coefficients of its Poincaré polynomial have many interpretations. An interesting one is provided by the Varchenko-Gel’fand ring, which is the ring of functions from the chambers of the arrangement to the integers with pointwise addition and multiplication. Varchenko and Gel’fand gave a simple presentation for this ring, along with a filtration and associated graded ring whose Hilbert series is the Poincaré polynomial. We generalize these results to cones defined by intersections of halfspaces of some of the hyperplanes and prove a novel result for the Varchenko-Gel’fand ring of an arrangement: when the arrangement is supersolvable the associated graded ring of the arrangement is Koszul.
MSC:
| 52C35 | Arrangements of points, flats, hyperplanes (aspects of discrete geometry) |
| 05Axx | Enumerative combinatorics |
| 52C40 | Oriented matroids in discrete geometry |
Software:
SageMathReferences:
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