A second reduction-type formula for the refined Littlewood--Richardson coefficients in type A

2026/07/26 by Siddheswar Kundu
Mathematics · #math.RT

paper · pdf

Abstract

For a permutation w in the symmetric group S_n and partitions λ, μ, ν with at most n parts, the refined Littlewood--Richardson (LR) coefficients c^ν_λ,μ(w) in type A_n count the multiplicity of the irreducible polynomial representation V(ν) of the general linear algebra \mathfrakgl_n(ℂ) appearing in the decomposition of the Kostant--Kumar submodule K(λ,w,μ) of the tensor product V(λ) ⊗ V(μ) of two irreducible polynomial \mathfrakgl_n(ℂ)-modules. In this paper, we establish a second reduction-type formula for c^ν_λ,μ(w), extending the second reduction formula for the classical Littlewood--Richardson coefficients c^ν_λ,μ. The proof relies on the hive model.

Citations

• The saturation property for refined Littlewood-Richardson coefficients
• Schubert calculus from polyhedral parametrizations of Demazure crystals
• A study of Kostant-Kumar modules via Littelmann paths
• The saturation conjecture (after A. Knutson and T. Tao)

Related

• Some unexpected properties of Littlewood-Richardson coefficients
• Contingency tables and the generalized Littlewood-Richardson coefficients
• Bidilatation of Small Littlewood-Richardson Coefficients
• The saturation property for refined Littlewood-Richardson coefficients
• Multiplication of polynomials on Hermitian symmetric spaces and Littlewood-Richardson coefficients