کد مقاله | کد نشریه | سال انتشار | مقاله انگلیسی | نسخه تمام متن |
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4654237 | 1632817 | 2009 | 31 صفحه PDF | دانلود رایگان |
A new isomorphism invariant of matroids is introduced, in the form of a quasisymmetric function. This invariant:
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• defines a Hopf morphism from the Hopf algebra of matroids to the quasisymmetric functions, which is surjective if one uses rational coefficients;
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• is a multivariate generating function for integer weight vectors that give minimum total weight to a unique base of the matroid;
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• is equivalent, via the Hopf antipode, to a generating function for integer weight vectors which keeps track of how many bases minimize the total weight;
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• behaves simply under matroid duality;
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• has a simple expansion in terms of PP-partition enumerators;
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• is a valuation on decompositions of matroid base polytopes. This last property leads to an interesting application: it can sometimes be used to prove that a matroid base polytope has no decompositions into smaller matroid base polytopes. Existence of such decompositions is a subtle issue arising from the work of Lafforgue, where lack of such a decomposition implies that the matroid has only a finite number of realizations up to scalings of vectors and overall change-of-basis.
Journal: European Journal of Combinatorics - Volume 30, Issue 8, November 2009, Pages 1727–1757