Alcove walk models for parabolic Mirkovi\'c-Vilonen intersections and branching to Levi subgroups

TL;DR

This paper introduces alcove walk models for parabolic Mirković-Vilonen intersections in the affine Grassmannian, providing combinatorial tools for understanding their structure and branching rules in representation theory.

Contribution

It develops explicit alcove walk models for these intersections, offering a new combinatorial approach to branching and character computation in representation theory.

Findings

Explicit cellular pavings indexed by alcove walks

Parametrization of irreducible components via alcove walks

New algorithm for computing characters of highest weight representations

Abstract

This article establishes alcove walk models for intersections of Schubert varieties and partially semi-infinite orbits in the affine Grassmannian of a split reductive group (we call such intersections parabolic Mirkovi\'c-Vilonen intersections). More precisely, we describe explicit cellular pavings of these intersections, indexed by certain positively-folded alcove walks. We prove a parametrization of the irreducible components of maximal possible dimension, in terms of alcove walks of maximal possible dimension. We then deduce a new combinatorial description of branching to Levi subgroups of irreducible highest weight representations, and in particular we give a new algorithm for computing the characters of such representations.