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Polymatroids and finite groups

Ed Swartz, Cornell University
Wednesday, October 21, 2020 - 3:30pm to 5:00pm
via Zoom
Ed Swartz

Note: This talk begins with a pre-seminar (aimed at graduate students) at 3:30–4:00. The main talk starts at 4:10.

Join Zoom Meeting:
Meeting ID: 950 2088 4635

Abstract: The speaker will try to convince you that the analogy “Polymatroids are to finite groups as matroids are to finite fields” is nowhere near as crazy as it sounds. 

Intro Material:

  • MatroidsWhitney’s original 1935 paper, “On the abstract properties of linear independence” is highly relevant.  In fact, I will be telling a portion of the story in that paper.   An alternative is something like the first three sections of Oxley’s survey paper, “What is a matroid?"  In this case, one should add the definition of the rank function of the dual matroid which is not in Oxley's paper.  This can just be searched for on the web with something like “rank function of a dual matroid” or something similar.

  • Characteristic polynomial of a matroid: Zaslavsky’s chapter in “Combinatorial Geometries” edited by Neil White is a good reference.  All that is really needed is Proposition 7.2.1 (which can be taken as a definition of the characteristic polynomial of a matroid) and Theorem 7.6.1 (Critical Theorem of Crapo and Rota)

  • Representations of finite groups:  The following facts/concepts will come up:
    • Irreducible representations of finite groups (with complex coefficients) (What are they? There are only finite many.
    • Every representation is a direct sum of such representations in a sort of unique way. 
    • Representations of abelian groups - especially Z/pZ, p a prime
    • Irreducible representation of products of groups via tensor products.

These topics are in most intro to representation courses/ intro to algebra courses.  Serre’s “Linear representations of finite groups" has all of them in the first three chapters (Chapter one, sections 1.1-1.5, chapter 3). 

I will discuss some unknown subset of these topics in the pre-seminar and will also talk about them in the main talk, but much quicker.