On the Shifted Littlewood-Richardson Coefficients and Littlewood-Richardson Coefficients
Khanh Nguyen Duc
Abstract
We give a new interpretation of the shifted Littlewood-Richardson coefficients $f_{λμ}^ν$ ($λ,μ,ν$ are strict partitions). The coefficients $g_{λμ}$ which appear in the decomposition of Schur $Q$-function $Q_λ$ into the sum of Schur functions $Q_λ= 2^{l(λ)}\sum_μg_{λμ}s_μ$ can be considered as a special case of $f_{λμ}^ν$ (here $λ$ is a strict partition of length $l(λ)$). We also give another description for $g_{λμ}$ as the cardinal of a subset of a set that counts Littlewood-Richardson coefficients $c_{μ^tμ}^{\tildeλ}$. This new point of view allows us to establish connections between $g_{λμ}$ and $c_{μ^t μ}^{\tildeλ}$. More precisely, we prove that $g_{λμ}=g_{λμ^t}$, and $g_{λμ} \leq c_{μ^tμ}^{\tildeλ}$. We conjecture that $g_{λμ}^2 \leq c^{\tildeλ}_{μ^tμ}$ and formulate some conjectures on our combinatorial models which would imply this inequality if it is valid.