Generalized Littlewood-Richardson coefficients for branching rules of GL(n) and extremal weight crystals

2018/01/28 by Brett M. Collins, Brett Collins, Collins, Brett · 1 citation
Mathematics · #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #Algebraic structures and combinatorial models #Combinatorics (math.CO) #FOS: Mathematics #Representation Theory (math.RT) #math.CO #math.RT

paper · pdf · doi:10.48550/arxiv.1801.09170

28 pages, comments welcome

openalex publication_date 2018/01/28 · arxiv created 2018/11/15 · arxiv updated 2018/11/16 · openalex created_date 2022/10/04 · openalex updated_date 2026/07/28

Abstract

Following the methods used by Derksen-Weyman in \citeDW11 and Chindris in \citeChi08, we use quiver theory to represent the generalized Littlewood-Richardson coefficients for the branching rule for the diagonal embedding of \gl(n) as the dimension of a weight space of semi-invariants. Using this, we prove their saturation and investigate when they are nonzero. We also show that for certain partitions the associated stretched polynomials satisfy the same conjectures as single Littlewood-Richardson coefficients. We then provide a polytopal description of this multiplicity and show that its positivity may be computed in strongly polynomial time. Finally, we remark that similar results hold for certain other generalized Littlewood-Richardson coefficients.

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• Jacobi polynomials on the simplex of Hermitian matrices and zonal spherical functions on partial flag manifolds

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