[Paper Review] MV-cycles and MV-polytopes in type A

Jared E. Anderson, M. G. Kogan|arXiv (Cornell University)|May 21, 2003

Advanced Combinatorial Mathematics7 references2 citations

TL;DR

This paper studies MV-cycles in type A Lie algebras by partitioning the loop Grassmannian into smooth, coweight-invariant pieces whose closures are the MV-cycles. Using the lattice model, it explicitly describes points in each piece and computes the moment map images (MV-polytopes) by identifying their vertices, providing a complete combinatorial description of these polytopes via the Kostant parameter set.

ABSTRACT

Abstract. We study, in type A, the algebraic cycles (MV-cycles) discovered by I. Mirković and K. Vilonen [MV]. In particular, we partition the loop Grassmannian into smooth pieces such that the MV-cycles are their closures. We explicitly describe the points in each piece using the lattice model of the loop Grassmannian in type A. The partition is invariant under the action of the coweights and, up to this action, the pieces are parametrized by the Kostant parameter set. We compute the moment map images of MV-cycles (MV-polytopes) by identifying the vertices of each polytope. 1.

Motivation & Objective

• To understand the structure of MV-cycles in type A by analyzing their geometric and combinatorial properties.
• To partition the loop Grassmannian into smooth, coweight-invariant pieces whose closures are MV-cycles.
• To explicitly describe the points in each piece using the lattice model of the loop Grassmannian in type A.
• To compute the moment map images of MV-cycles (MV-polytopes) by identifying their vertices.
• To show that, up to coweight action, the pieces are parametrized by the Kostant parameter set.

Proposed method

• Partitioning the loop Grassmannian into smooth, coweight-invariant pieces using the lattice model in type A.
• Using the lattice model to explicitly describe the points in each piece of the partition.
• Analyzing the action of coweights on the partition to establish invariance and parametrization.
• Identifying the vertices of MV-polytopes via the moment map image computation.
• Relating the parametrization of pieces to the Kostant parameter set under the coweight action.
• Establishing a combinatorial correspondence between MV-cycles and vertices of MV-polytopes.

Experimental results

Research questions

• RQ1 How can the loop Grassmannian in type A be partitioned into smooth, coweight-invariant pieces whose closures are MV-cycles?
• RQ2 What is the explicit description of points in each piece of the partition using the lattice model?
• RQ3 How do the moment map images of MV-cycles (MV-polytopes) relate to their vertex structures?
• RQ4 In what way is the parametrization of the pieces related to the Kostant parameter set, up to coweight action?
• RQ5 What combinatorial structure underlies the vertices of MV-polytopes in type A?

Key findings

• The loop Grassmannian in type A is partitioned into smooth, coweight-invariant pieces whose closures are MV-cycles.
• Each piece in the partition is explicitly described using the lattice model of the loop Grassmannian.
• The vertices of MV-polytopes are fully determined by identifying the moment map images of the MV-cycles.
• Up to the action of coweights, the pieces of the partition are parametrized by the Kostant parameter set.
• The combinatorial structure of MV-polytopes in type A is completely characterized by their vertices, which are computed via the moment map.
• The study establishes a precise link between the geometry of MV-cycles and their polytopal images through vertex identification.