On the Shifted Littlewood–Richardson Coefficients and the Littlewood–Richardson Coefficients

Abstract

We give a new interpretation of the shifted Littlewood–Richardson coefficients $f_{\lambda \mu }^\nu$ ($\lambda ,\mu ,\nu$ are strict partitions). The coefficients $g_{\lambda \mu }$ which appear in the decomposition of Schur Q-function $Q_\lambda$ into the sum of Schur functions $Q_\lambda = 2^{l(\lambda )}\sum \nolimits _{\mu }g_{\lambda \mu }s_\mu$ can be considered as a special case of $f_{\lambda \mu }^\nu$ (here $\lambda$ is a strict partition of length $l(\lambda )$). We also give another description for $g_{\lambda \mu }$ as the cardinal of a subset of a set that counts Littlewood–Richardson coefficients $c_{\mu ^t\mu }^{{\tilde{\lambda }}}$. This new point of view allows us to establish connections between $g_{\lambda \mu }$ and $c_{\mu ^t \mu }^{{\tilde{\lambda }}}$. More precisely, we prove that $g_{\lambda \mu }=g_{\lambda \mu ^t}$, and $g_{\lambda \mu } \le c_{\mu ^t\mu }^{{\tilde{\lambda }}}$. We conjecture that $g_{\lambda \mu }^2 \le c^{{\tilde{\lambda }}}_{\mu ^t\mu }$ and formulate some conjectures on our combinatorial models which would imply this inequality if it is valid.

Appendix

The construction of the set $\widetilde{\mathcal {O}}(\nu /\mu )$ in the Example 4.5