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A Grassmann algebra for matroids

Jeffrey Giansiracusa Orcid Logo, Noah Giansiracusa

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

Swansea University Author: Jeffrey Giansiracusa Orcid Logo

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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...

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Published in: manuscripta mathematica
ISSN: 0025-2611 1432-1785
Published: 2018
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URI: https://cronfa.swan.ac.uk/Record/cronfa33671
first_indexed 2017-05-15T19:03:03Z
last_indexed 2020-07-08T12:52:17Z
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spelling 2020-07-08T11:50:20.2316765 v2 33671 2017-05-15 A Grassmann algebra for matroids 03c4f93e1b94af60eb0c18c892b0c1d9 0000-0003-4252-0058 Jeffrey Giansiracusa Jeffrey Giansiracusa true false 2017-05-15 MACS 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. Journal Article manuscripta mathematica 156 1-2 187 213 0025-2611 1432-1785 matroid, exterior algebra, tropical geometry, idempotent algebra, semiring 31 12 2018 2018-12-31 10.1007/s00229-017-0958-z COLLEGE NANME Mathematics and Computer Science School COLLEGE CODE MACS Swansea University 2020-07-08T11:50:20.2316765 2017-05-15T17:07:21.1655219 Faculty of Science and Engineering School of Mathematics and Computer Science - Mathematics Jeffrey Giansiracusa 0000-0003-4252-0058 1 Noah Giansiracusa 2 0033671-15052017170809.pdf Grassmann.pdf 2017-05-15T17:08:09.7430000 Output 241567 application/pdf Accepted Manuscript true 2018-07-26T00:00:00.0000000 true eng
title A Grassmann algebra for matroids
spellingShingle A Grassmann algebra for matroids
Jeffrey Giansiracusa
title_short A Grassmann algebra for matroids
title_full A Grassmann algebra for matroids
title_fullStr A Grassmann algebra for matroids
title_full_unstemmed A Grassmann algebra for matroids
title_sort A Grassmann algebra for matroids
author_id_str_mv 03c4f93e1b94af60eb0c18c892b0c1d9
author_id_fullname_str_mv 03c4f93e1b94af60eb0c18c892b0c1d9_***_Jeffrey Giansiracusa
author Jeffrey Giansiracusa
author2 Jeffrey Giansiracusa
Noah Giansiracusa
format Journal article
container_title manuscripta mathematica
container_volume 156
container_issue 1-2
container_start_page 187
publishDate 2018
institution Swansea University
issn 0025-2611
1432-1785
doi_str_mv 10.1007/s00229-017-0958-z
college_str Faculty of Science and Engineering
hierarchytype
hierarchy_top_id facultyofscienceandengineering
hierarchy_top_title Faculty of Science and Engineering
hierarchy_parent_id facultyofscienceandengineering
hierarchy_parent_title Faculty of Science and Engineering
department_str School of Mathematics and Computer Science - Mathematics{{{_:::_}}}Faculty of Science and Engineering{{{_:::_}}}School of Mathematics and Computer Science - Mathematics
document_store_str 1
active_str 0
description 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.
published_date 2018-12-31T19:08:16Z
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score 11.04748

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