Alcove walks, buildings, symmetric functions and representations

2008/07/23 by James Parkinson, Arun Ram, Parkinson, James +1 · 1 citation
Computer Science · Mathematics · #20E42 #20G05 #Advanced Combinatorial Mathematics #Combinatorics (math.CO) #Computational Geometry and Mesh Generation #FOS: Mathematics #Representation Theory (math.RT) #Topological and Geometric Data Analysis

paper · pdf · doi:10.48550/arxiv.0807.3602

openalex publication_date 2008/07/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

For a complex simple Lie algebra, the dimension K_λμ of the μ weight space of a finite dimensional representation of highest weight λ is the same as the number of Littelmann paths of type λ and weight μ. In this paper we give an explicit construction of a path of type λ and weight μ whenever K_λμ≠ 0. This construction has additional consequences, it produces an explicit point in the building which chamber retracts to λ and sector retracts to μ, and an explicit point of the affine Grassmannian in the corresponding Mirković-Vilonen intersection. In an appendix we discuss the connection between retractions in buildings and alcove walks.

Citations

• A polytope combinatorics for semisimple groups
• Mirkovic-Vilonen cycles and polytopes

Cited by

• Kostant Convexity for affine buildings
• Non-discrete affine buildings and convexity

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