Threefold Schur coefficients $c_{a,b,c}^{\delta }$

For partitions $\delta ,a,b,c$ with $\mathrm{\mid }\delta \mathrm{\mid }=\mathrm{\mid }a\mathrm{\mid }+\mathrm{\mid }b\mathrm{\mid }+\mathrm{\mid }c\mathrm{\mid }$, define the threefold Schur coefficient $c_{a,b,c}^{\delta }:=[s_{\delta }]s_a{s}_b{s}_c,$ the coefficient of $s_{\delta }$ in the ordinary product of the three Schur functions of $\mathrm{\Lambda }$ (Stable Schur functions from bialternants); when $\mathrm{\mid }\delta \mathrm{\mid }\mathrm{\ne }\mathrm{\mid }a\mathrm{\mid }+\mathrm{\mid }b\mathrm{\mid }+\mathrm{\mid }c\mathrm{\mid }$ the coefficient is $0$, the product being homogeneous of degree $\mathrm{\mid }a\mathrm{\mid }+\mathrm{\mid }b\mathrm{\mid }+\mathrm{\mid }c\mathrm{\mid }$. Since $s_a{s}_b={\sum }_{\varphi }{c}_{a,b}^{\varphi }{s}_{\varphi }$ with $c_{a,b}^{\varphi }$ the Littlewood–Richardson coefficients (The Littlewood–Richardson rule for products of Schur functions, Littlewood--Richardson tableaux and coefficients), substituting gives $c_{a,b,c}^{\delta }={\sum }_{\varphi }{c}_{a,b}^{\varphi }{c}_{\varphi ,c}^{\delta }={\sum }_{\psi }{c}_{a,c}^{\psi }{c}_{\psi ,b}^{\delta },$ the sums over partitions $\varphi$ of $\mathrm{\mid }a\mathrm{\mid }+\mathrm{\mid }b\mathrm{\mid }$ and $\psi$ of $\mathrm{\mid }a\mathrm{\mid }+\mathrm{\mid }c\mathrm{\mid }$; hence each $c_{a,b,c}^{\delta }$ is a nonnegative integer, and the two displayed expressions agree with the definition. These coefficients are symmetric in $a,b,c$ because the product $s_a{s}_b{s}_c$ is commutative, and equivalently $c_{a,b,c}^{\delta }=⟨s_a{s}_b{s}_c,s_{\delta }{⟩}_H,$ by the orthonormality of the Schur basis (Schur functions form an orthonormal integral basis). No fourfold or longer iterated coefficient is introduced here; iterating the two identities above gives every nested product of three factors, and the symmetry of the three subscripts is used throughout this page.