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A path model for MV polytopes in type A_n

Zijun Li

Abstract

We introduce a one-skeleton path model for Mirkovic-Vilonen polytopes in type A_n. We prove that the Minkowski sum of (MV) polytopes corresponds to the concatenation of one-skeleton paths of this model. We show that MV polytopes induced by fundamental one-skeleton paths are Harder-Narasimhan polytopes. The paths given by an orientation of the fundamental alcove parameterize precisely the cluster variables in the initial seed of the coordinate ring C[N]. We also establish a correspondence between fundamental one-skeleton paths and folded galleries representing maximal faces of subword complexes. Under this correspondence, the comultiplication structure of C[N] matches the intrinsic comultiplication structure of folded galleries given by projections to sub-Coxeter complexes.

A path model for MV polytopes in type A_n

Abstract

We introduce a one-skeleton path model for Mirkovic-Vilonen polytopes in type A_n. We prove that the Minkowski sum of (MV) polytopes corresponds to the concatenation of one-skeleton paths of this model. We show that MV polytopes induced by fundamental one-skeleton paths are Harder-Narasimhan polytopes. The paths given by an orientation of the fundamental alcove parameterize precisely the cluster variables in the initial seed of the coordinate ring C[N]. We also establish a correspondence between fundamental one-skeleton paths and folded galleries representing maximal faces of subword complexes. Under this correspondence, the comultiplication structure of C[N] matches the intrinsic comultiplication structure of folded galleries given by projections to sub-Coxeter complexes.

Paper Structure

This paper contains 26 sections, 30 theorems, 119 equations.

Table of Contents

  1. Introduction
  2. Crystals and perfect bases
  3. MV polytopes and path models
  4. Subword complexes, proto-exact categories and Hall algebras
  5. Outline
  6. Preliminaries
  7. Perfect basis and crystals
  8. Mirković-Vilonen polytopes
  9. Subword complexes
  10. One-skeleton path model
  11. One-skeleton paths
  12. Polytopal map
  13. The free path algebra Fp
  14. MV polytopes from fundamental one-skeleton paths
  15. Crystal graphs for fundamental one-skeleton paths
  16. ...and 11 more sections

Key Result

Theorem 1

For two one-skeleton paths $p$ and $q$, we have where $\operatorname{Pol}$ is a map from one-skeleton paths to polytopes, $*$ is the concatenation of paths and $\boxplus$ is the Minkowski sum of polytopes.

Theorems & Definitions (82)

  • Theorem
  • Definition 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Definition 3.1
  • Definition 3.2
  • ...and 72 more