Schur Duality and Strict Polynomial Functors

Nico Courts

Masters thesis, 2020

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Abstract:

We begin by going through a considerable amount of domain knowledge concerning representations of $\operatorname{GL}_n$, representations of $\mathfrak S_n$, tracing the development from the classical study of group representations by Schur and Weyl and the transformation of this theory in the more robust language of affine group schemes. From there, the story takes on a more categorical flavor as we discuss different manifestations of polynomial representations of $\operatorname{GL}_n$, following the work of Friedlander and Suslin as well as Krause, Aquilino, and Reischuk. In the latter case, we show how they determined that the Schur-Weyl functor is monoidal, opening up the theory to the machinery of monoidal categories. We take some time to develop the theory of tensor triangulated geometry from Balmer as well as discuss some standard constructions necessary for the theory. We end our paper by talking about how all of this work comes together to elucidate some problems at the boundaries of modern representation theory as well as techniques with which we can solve them.

Bibtex:

@mastersthesis{courts2020mastersthesis,
  title={Schur Duality and Strict Polynomial Functors},
  author={Courts, Nicolas},
  year={2020}
}