Bidilatation of Small Littlewood-Richardson Coefficients

2022/06/07 by Chaput, Pierre-Emmanuel, Ressayre, Nicolas
#Algebraic Geometry (math.AG) #FOS: Mathematics #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.2206.03054

Abstract

The Littlewood-Richardson coefficients c^ν_λ,μ are the multiplicities in the tensor product decomposition of two irreducible representations of the general linear group GL(n, \mathbb C). They are parametrized by the triples of partitions (λ, μ, ν) of length at most n. By the so-called Fulton conjecture, if c^ν_λ,μ=1 then c^kν_kλ,kμ= 1, for any k ≥ 0. Similarly, as proved by Ikenmeyer or Sherman, if c^ν_λ,μ=2 then c^kν_kλ,kμ = k + 1, for any k≥ 0. Here, given a partition λ, we set λ(p, q) = p(qλ')' , where prime denotes the conjugate partition. We observe that Fulton's conjecture implies that if c^ν_λ,μ=1 then c^ν(p,q)_{λ(p,q),μ(p,q)}=1, for any p, q ≥ 0. Our main result is that if c^ν_λ,μ=2 then c^ν(p,q)_{λ(p,q),μ(p,q)} is the binomial \beginpmatrix p+q q \endpmatrix, for any p, q ≥ 0.

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