A modification of the Anderson-Mirkovic conjecture for Mirkovic-Vilonen polytopes in types B and C

TL;DR

This paper provides an explicit method for applying Kashiwara operators to Mirković-Vilonen polytopes in types B and C, modifying the Anderson-Mirković conjecture and enabling inductive generation of these polytopes.

Contribution

It offers a new explicit description of Kashiwara operators for MV polytopes in types B and C, extending previous work and addressing conjecture modifications.

Findings

Explicit description of Kashiwara operators in types B and C

Inductive generation method for MV polytopes in these types

Counterexample to original conjecture in type C_{3}

Abstract

We give an explicit description of the (lowering) Kashiwara operators on Mirkovi\'c-Vilonen polytopes in types $B$ and $C$, which provides a simple method for generating Mirkovi\'c-Vilonen polytopes inductively. This description can be thought of as a modification of the original Anderson-Mirkovi\'c conjecture, which Kamnitzer proved in the case of type $A$, and presented a counterexample in the case of type $C_{3}$.