Representation theory of orthogonal matroids
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Jin, Tong
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Abstract
In this dissertation, we study the representation theory of orthogonal matroids.
The introduction contains the motivation for the dissertation. We review the representation theory of matroids, with an emphasis on the key ingredients that are waiting for development in the context of orthogonal matroids.
After quickly recalling the established theory of orthogonal matroids at the beginning of Chapter 2, we define and study basic properties of the extended rank function and the modular tuples in orthogonal matroids. We then prove a weak version of the path theorem on the set of circuits of an orthogonal matroid.
In Chapter 3, we consider representations of orthogonal matroids over tracts by bases. We then give a few applications, purely using this basis approach, to the representation theory of orthogonal matroids. In Chapter 4, we give a different way of representing orthogonal matroids by circuit functions, which is proved to be equivalent to the basis approach. This is based on joint work with Matthew Baker and joint work with Donggyu Kim.
The final chapter focuses on the rescaling classes of representations. We construct the foundation of an orthogonal matroid, which possesses the universal property that the set of rescaling classes of representations is in one-to-one correspondence with the set of morphisms from the foundation to the target pasture. We also give explicit generators and relations of the foundation, along with an algorithm for computations.
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2025-07-23
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Dissertation