MV polytopes and reduced double Bruhat cells

  • Kathlyn Dykes

    University of Toronto, Toronto, Canada
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Abstract

When is a complex reductive algebraic group, MV polytopes are in bijection with the non-negative tropical points of the unipotent group of . By fixing from the Weyl group, we can define MV polytopes whose highest vertex is labelled by . We show that these polytopes are in bijection with the non-negative tropical points of the reduced double Bruhat cell labelled by . To do this, we define a collection of generalized minor functions which tropicalize on the reduced Bruhat cell to the BZ data of an MV polytope of highest vertex . We also describe the combinatorial structure of MV polytopes of highest vertex . We explicitly describe the map from the Weyl group to the subset of elements bounded by in the Bruhat order which sends if the vertex labelled by coincides with the vertex labelled by for every MV polytope of highest vertex . As a consequence of this map, we prove that these polytopes have vertices labelled by Weyl group elements less than in the Bruhat order.

Cite this article

Kathlyn Dykes, MV polytopes and reduced double Bruhat cells. J. Comb. Algebra (2024), published online first

DOI 10.4171/JCA/86