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The Varchenko Matrix for Cones
Consider an arrangement of hyperplanes and assign to each hyperplane a weight. By using this weights Varchenko defines a bilinear form on the vector space freely generated by the regions of the arrangement. We define this bilinear form for cones of the arrangement. Then we show that the determinant...
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| Format: | Dissertation |
| Sprache: | Englisch |
| Veröffentlicht: |
Philipps-Universität Marburg
2013
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| Online Zugang: | PDF-Volltext |
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| Zusammenfassung: | Consider an arrangement of hyperplanes and assign to each hyperplane a weight. By using this weights Varchenko defines a bilinear form on the vector space freely generated by the regions of the arrangement. We define this bilinear form for cones of the arrangement. Then we show that the determinant of the matrix of the bilinear form restricted to the cone is determined by the combinatorics of the arrangement inside the cone and factors nicely. The resulting theorem induces Varchenko's thereom about the determinant of the matrix of the Varchenko bilinear form.
We consider cones of the braid arrangement which are defined by partially ordered sets. We give a formular for the determinant of a matrices of the bilinear form restricted to one this cones by using properties of the partially ordered set. |
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| DOI: | 10.17192/z2013.0480 |