Heilbronn Algebra Seminar - Ben Smith
| Dates: | 7 October 2025 |
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| Times: | 14:00 - 15:00 |
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| What is it: | Seminar |
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| Organiser: | Department of Mathematics |
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| Who is it for: | University staff, External researchers, Current University students |
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| Speaker: | Ben Smith |
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Title: Coxeter matroids, minuscule varieties and Plücker equations
Abstract: A matroid is a combinatorial object that can be seen as an abstraction of a linear space. For example, it is defined via a basis exchange relation, and every linear space gives rise to an underlying matroid. However, there is a deeper connection via the Grassmannian, the variety of subspaces of a vector space cut out by the Plücker equations. Explicitly, the space of matroids is a ’combinatorial Grassmannian’ cut out by the Plücker equations over the Boolean semifield. The process that gives these combinatorial equations is called tropicalisation. Grassmannians and matroids are inherently ’type A’ objects, and both have abstractions associated to other root systems. Grassmannians are particular types of miniscule varieties. As such, one can generalise to quotients G/P of a simply connected complex Lie group G by a maximal parabolic subgroup P with a minuscule fundamental representation. For matroids, one can generalise to Coxeter matroids whose exchange relations are governed by different root systems. We generalise the connection between these two paradigms to other types, showing that certain spaces of Coxeter matroids are cut out by the tropicalisation of the equations that define G/P. (This is joint work with Aram Dermenjian and Alex Fink and Kieran Calvert).
Speaker
Ben Smith
Organisation: University of Lancaster
Travel and Contact Information
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Frank Adams 1
Alan Turing Building
Manchester
