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A Grassmann algebra for matroids / Jeffrey Giansiracusa; Noah Giansiracusa

manuscripta mathematica, Volume: 156, Issue: 1-2, Pages: 187 - 213

Swansea University Author: Giansiracusa, Jeffrey

Abstract

We introduce an idempotent analogue of the exterior algebra for which the theory of tropical linear spaces (and valuated matroids) can be seen in close analogy with the classical Grassmann algebra formalism for linear spaces. The top wedge power of a tropical linear space is its Plucker vector, whic...

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Published in: manuscripta mathematica
ISSN: 0025-2611 1432-1785
Published: 2018
Online Access: Check full text

URI: https://cronfa.swan.ac.uk/Record/cronfa33671
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Abstract: We introduce an idempotent analogue of the exterior algebra for which the theory of tropical linear spaces (and valuated matroids) can be seen in close analogy with the classical Grassmann algebra formalism for linear spaces. The top wedge power of a tropical linear space is its Plucker vector, which we view as a tensor, and a tropical linear space is recovered from its Plucker vector as the kernel of the corresponding wedge multiplication map. We prove that an arbitrary d-tensor satisfies the tropical Plucker relations (valuated exchange axiom) if and only if the d-th wedge power of the kernel of wedge-multiplication is free of rank one. This provides a new cryptomorphism for valuated matroids, including ordinary matroids as a special case.
Keywords: matroid, exterior algebra, tropical geometry, idempotent algebra, semiring
College: College of Science
Issue: 1-2
Start Page: 187
End Page: 213