Crystal Combinatorics and Geometric Satake

TL;DR

This paper surveys various combinatorial models for MV cycles, including polytopes, algebras, crystals, and galleries, providing explicit examples to clarify their interrelations.

Contribution

It compiles and compares multiple combinatorial models for MV cycles, highlighting their connections and providing explicit computations.

Findings

Identified key combinatorial models for MV cycles

Explicitly computed examples illustrating model connections

Clarified relationships between different combinatorial approaches

Abstract

This is an REU paper written for the University of Chicago REU, summer 2017. The main purpose of this note is to collect some of the many combinatorial models for MV cycles that exist in the literature. In particular, we will investigate MV polytopes, preprojective algebras, crystals, and LS galleries. We also compute explicitly some examples to help elucidate the theory and the connections between these models.