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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: Jeffrey Giansiracusa
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DOI (Published version): 10.1007/s00229-017-0958-z
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...
| Published in: | manuscripta mathematica |
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| ISSN: | 0025-2611 1432-1785 |
| Published: |
2018
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| Online Access: |
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| 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. |
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| Keywords: |
matroid, exterior algebra, tropical geometry, idempotent algebra, semiring |
| College: |
Faculty of Science and Engineering |
| Issue: |
1-2 |
| Start Page: |
187 |
| End Page: |
213 |