## Departmental PhD Thesis Exam – Kathlyn Dykes

Thursday, August 18, 2022 at 10:00 a.m. (sharp)

PhD Candidate: Kathlyn Dykes
Supervisor: Joel Kamnitzer
Thesis title: MV polytopes and reduced double Bruhat cells

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When $G$ is a complex reductive algebraic group, MV polytopes are in bijection with the non-negative tropical points of the unipotent group of $G$. By fixing $w$ from the Weyl group, we can define MV polytopes whose highest vertex is labelled by $w$. We show that these polytopes are in bijection with the non-negative tropical points of the reduced double Bruhat cell labelled by $w^{-1}$. To do this, we define a collection of generalized minor functions $\Delta_\gamma^\text{new}$ which tropicalize on the reduced Bruhat cell to the BZ data of an MV polytope of highest vertex $w$.

We also describe the combinatorial structure of MV polytopes of highest vertex $w$. We explicitly describe the map from the Weyl group to the subset of elements bounded by $w$ in the Bruhat order which sends $u \mapsto v$ if the vertex labelled by $u$ coincides with the vertex labelled by $v$ for every MV polytope of highest vertex $w$. As a consequence of this map, we prove that these polytopes have vertices labelled by Weyl group elements less than $w$ in the Bruhat order.

A motivation for studying MV polytopes of highest vertex $w$ is that they are the finite-type equivalent of lower affine MV polytopes for $\widehat{SL_2}$. We show that for $\ell(w) \leq 3$, lower affine MV polytopes with highest vertex $w$ are in bijection with the non-negative tropical points of the reduced double Bruhat cell labelled by $w^{-1}$ for $\widehat{SL_2}$.

Finally, MV polytopes in the finite case are defined by the tropical Pl\”{u} relations while rank 2 affine MV polytopes are defined by “diagonal relations”. We prove that for $B_2$ polytopes, these diagonal relations hold and are equivalent to the tropical Pl\”{u}cker relations.

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A copy of the thesis can be found here: Dykes_Kathlyn_thesis