is there an "elementary" (read: short combinatorial) proof for the rule $$ s_\lambda \cdot s_{(1)} = \sum_{\mu} s_{\mu} $$ where $\mu$ ranges over all partitions obtained from $\lambda$ by adding a singe box?
Where the Schur-polynomial $s_\lambda$ is defined as the sum over all monomials one gets from semi-standard tableaux corresponding to the Young-diagram of $\lambda$. The number of variables here is $n$ and $\lambda$ is a partition of $d\leq n-1$.
Basically, I would like to understand Schur-Weyl duality, that is, the decomposition of the d-fold tensor product as described in chapter 6 of Fulton, Harris "Representation theory". So alternatively, if $W(\lambda)$ if the irreducible representation of $GL_n(\mathbb{C})$ corresponding to $\lambda$, then I want to understand the decomposition of $W(\lambda) \otimes \mathbb{C}^n$.
$\dim W(\lambda)$ is the number of semistandard tableaux filled with numbers from $[n]$. The identity to be proved would then be $$ \dim W(\lambda) \cdot n = \sum_{\mu} \dim W(\mu) $$