Positroids, non-crossing partitions, and positively oriented matroidsReportar como inadecuado

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1 San Francisco State University - Department of Mathematics 2 Universidad de los Andes Bogota 3 WMI - Warwick Mathematics Institute 4 Department of Mathematics Berkeley

Abstract : We investigate the role that non-crossing partitions play in the study of positroids, a class of matroids introduced by Postnikov. We prove that every positroid can be constructed uniquely by choosing a non-crossing partition on the ground set, and then freely placing the structure of a connected positroid on each of the blocks of the partition. We use this to enumerate connected positroids, and we prove that the probability that a positroid on n is connected equals $1-e^2$ asymptotically. We also prove da Silva-s 1987 conjecture that any positively oriented matroid is a positroid; that is, it can be realized by a set of vectors in a real vector space. It follows from this result that the positive matroid Grassmannian or positive MacPhersonian is homeomorphic to a closed ball.

Keywords : positroid non-crossing partition matroid polytope oriented matroid matroid Grassmannian.

Autor: Federico Ardila - Felipe Rincón - Lauren Williams -

Fuente: https://hal.archives-ouvertes.fr/


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