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Séminaire Lotharingien de Combinatoire, 85B.52 (2021), 12 pp.

# Mrigendra Singh Kushwaha, K. N. Raghavan and Sankaran Viswanath

# The saturation Problem for Refined Littlewood-Richardson Coefficients

**Abstract.**
Given partitions λ, μ, ν with at most *n* nonzero parts and a permutation *w* ∈ *S*_{n}, the refined Littlewood-Richardson coefficient *c*_{λμ}^{ν}(*w*) is the multiplicity of the irreducible *GL*_{n}**C** module *V*(ν) in the so-called Kostant-Kumar submodule *K*(λ,*w*,μ) of the tensor product *V*(λ) ⊗ *V*(μ). We derive a hive model for these coefficients and prove that the saturation property holds if *w* is 312-avoiding, 231-avoiding or a commuting product of such elements. This generalizes the classical Knutson-Tao saturation theorem.

Received: December 1, 2020.
Accepted: March 1, 2021.
Final version: April 29, 2021.

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