Lower and Upper Bounds for Nonzero Littlewood-Richardson Coefficients

TL;DR

This paper establishes bounds on partitions for which Littlewood-Richardson coefficients are positive, using tableau characterization and dominance order, aiding in understanding the structure of these coefficients.

Contribution

It introduces a method to compute lower and upper bounds for partitions with positive Littlewood-Richardson coefficients based on tableau counts and dominance order.

Findings

Derived bounds for partitions with positive coefficients

Algorithm based on tableau characterization and dominance order

Provides a systematic way to estimate nonzero coefficients

Abstract

Given a skew diagram $\gamma/\lambda$, we determine a set of lower and upper bounds that a partition $\mu$ must satisfy for Littlewood-Richards coefficients $c^{\gamma}_{\lambda,\mu}>0$. Our algorithm depends on the characterization of $c^{\gamma}_{\lambda,\mu}$ as the number of Littlewood-Richardson tableau of shape $\gamma/\lambda$ and content $\mu$ and uses the (generalized) dominance order on partitions as the main ingredient.