I am trying to compute Littlewood-Richardson coefficients involving hook-shaped diagrams. In particular, if $\lambda$ is a hook-shaped Young diagram, $\rho$ is any diagram, and we consider, $$ s_\lambda s_\rho = \sum_\nu c_{\lambda \rho}^\nu s_\nu $$ is there a simple expression of the sum over $\nu$ on the right?
The reason I suspect a simplification is because of the Pieri formulas, which state that for a single row $\lambda =(r) $ and $\rho $ arbitrary, $$ s_\rho s_{(r)} = s_\rho h_r=\sum_\kappa s_\kappa~, $$ where the sum is over all $\kappa$ so that $\kappa/\rho$ is a horizontal strip of weight $r$ and $h_k$ is the complete homogeneous symmetric polynomial. Similarly for $\lambda = (1^s)$ a column, $$ s_\rho s_{(1^s)} =s_\rho e_s= \sum_\kappa s_\kappa~, $$ where the sum is now over $\kappa$ so that $\kappa/\rho$ is a vertical strip of weight $s$ and $e_s$ is the elementary symmetric polynomial.
For a hook-shaped partition $(r,1^s)$, we have $$ s_{(r,1^s)}=\sum_{k=0}^s (-1)^k h_{r+k} e_{s-k}~, $$ so it is possible to calculate the product of a hook-shape with another diagram by applying the Pieri formulas to the above expansion of $s_{(r,1^s)}$. However, this will generally give a complicated expression with many terms cancelling out, so I suspect that a simpler expression exists. Any help is much appreciated.