Stretched Newell-Littlewood coefficients

Abstract

Newell-Littlewood coefficients $n_{\mu,\nu}^{\lambda}$ are the multiplicities occurring in the decomposition of products of universal characters of the orthogonal and symplectic groups. They may also be expressed, or even defined directly in terms of Littlewood-Richardson coefficients, $c_{\mu,\nu}^{\lambda}$. Both sets of coefficients have stretched forms $c_{t\mu,t\nu}^{t\lambda}$ and $n_{t\mu,t\nu}^{t\lambda}$, where $t\kappa$ is the partition obtained by multiplying each part of the partition $\kappa$ by the integer $t$. It is known that $c_{t\mu,t\nu}^{t\lambda}$ is a polynomial in $t$ and here it is shown that $n_{t\mu,t\nu}^{t\lambda}$ is an Ehrhart quasi-polynomial in $t$ with minimum quasi-period at most $2$. The evaluation of $n_{t\mu,t\nu}^{t\lambda}$ is effected both by deriving their generating function and by establishing a hive model analogous to that used for the calculation of $c_{t\mu,t\nu}^{t\lambda}$. These two approaches lead to a whole battery of conjectures about the nature of the quasi-polynomials $n_{t\mu,t\nu}^{t\lambda}$. These include both positivity, stability and saturation conjectures that are supported by a significant amount of data from a range of examples.