Geometry, Topology and Categorification Seminar joint with Combinatorics and Algebra Group: Representation theory for oriented matroids

Monday, March 22 at 2:30pm to 3:45pm

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Ethan Kowalenko, UC Riverside

Abstract: In the 2000's, Braden-Licata-Proudfoot-Webster (BLPW) defined and investigated an interesting finite-dimensional algebra $A$ associated to a \emph{polarized arrangement} $\mathcal{V}$, which is a real hyperplane arrangement with some extra data. The algebra $A$ shares many properties with category $\mathcal{O}$, a well-known category coming from the representation theory of complex semisimple Lie algebras. While representation theoretic in flavor, the properties of $A$ are proved using only the topology and combinatorics of $\mathcal{V}$.
The combinatorics and topology of real hyperplane arrangements are generalized by \emph{oriented matroids}. Every real hyperplane arrangement gives an oriented matroid, and every oriented matroid comes from a ``pseudophere arrangement.'' In this talk, we will discuss $A$ and its properties, and then consider a new generalization of $A$ to the world of oriented matroids. While our new algebras retain many basic features of $A$, some properties require an additional assumption on the input. In general, this assumption does not hold, and its failure could lead to a new invariant for hyperplane arrangements and directed graphs. This is joint work with Carl Mautner.

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Meeting ID: 996 9808 2212
Passcode: 102110

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Lecture / Talk / Workshop

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