Alcove walk models for parabolic Mirkovi\'c-Vilonen intersections and branching to Levi subgroups
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.
Keywords
Cite
@article{arxiv.2405.17174,
title = {Alcove walk models for parabolic Mirkovi\'c-Vilonen intersections and branching to Levi subgroups},
author = {Thomas J. Haines},
journal= {arXiv preprint arXiv:2405.17174},
year = {2026}
}
Comments
19 pages, 1 figure. Minor changes in wording made in introduction and in Proposition 3.2. Additional references added. Statement of Theorem A corrected, to account for a mistake in the previous version of Proposition 5.5. To appear, Transformation Groups; this arXiv version is close to but not the same as the Version of Record: https://doi.org/10.1007/s00031-026-09956-0