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arXiv · 2107.00981

Lift theorems for representations of matroids over pastures

Matthew Baker·Oliver Lorscheid
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Abstract

Pastures are a class of field-like algebraic objects which include both partial fields hyperfields and have nice categorical properties. We prove several lift theorems for representations of matroids over pastures, including a generalization of Pendavingh and van Zwam's Lift Theorem for partial fields. By embedding the earlier theory into a more general framework, we are able to establish new results even in the case of lifts of partial fields, for example the conjecture of Pendavingh--van Zwam that their lift construction is idempotent. We give numerous applications to matroid representations, e.g. we show that, up to projective equivalence, every pair consisting of a hexagonal representation and an orientation lifts uniquely to a near-regular representation. The proofs are different from the arguments used by Pendavingh and van Zwam, relying instead on a result of Gelfand-Rybnikov-Stone inspired by Tutte's homotopy theorem.

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Matthew Baker, Oliver Lorscheid. 2021-07-02. Lift theorems for representations of matroids over pastures. https://arxiv.org/abs/2107.00981

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