I am a physicist who has been studying (skew) Schur polynomials recently. I am particularly interested in skew Schur polynomials $s_{\lambda/\mu}$, where $\lambda$ is a representation corresponding to a hook-shaped Young diagram. Such skew Schur polynomials are of course only non-zero in the case where $\mu$ corresponds to a hook-shaped diagram as well.
For general representations $\lambda$ and $\mu$, skew Schur polynomials have rather complicated expressions. I suspect, however, that these expressions simplify in the case of representations corresponding to hook-shaped diagrams, but I have thus far failed to find evidence for this in the literature. I therefore have the following two questions:
Are there simple expressions for skew Schur polynomials $s_{\lambda/\mu}$, where $\lambda$ and $\mu$ correspond to hook-shaped diagrams?
Consider pairs of hook-shaped diagrams $\lambda,~\mu$ and $\kappa,~\nu$ such that $\lambda \neq \kappa$ or $\mu\neq \nu$ (i.e. $\lambda_i \neq \kappa_i$ for some $i$, where $\lambda_i$ is the length of row $i$ in the diagram of $\lambda$, similarly for $\mu_i$ and $\nu_i $). Are there such pairs of hook-shaped diagrams which satisfy $s_{\lambda/\mu} = s_{\kappa/\nu}$? It is known that $s_{\lambda/\mu}$ equals $s_{\lambda/\mu}=s_{({\lambda/\mu})^r}$, where $({\lambda/\mu})^r$ is obtained from ${\lambda/\mu}$ by rotating the skew shape by 180 degrees (see e.g. Stanley - Enumerative combinatorics, Exercise 7.56 a), but this does not give non-trivial identities for the case of hook-shaped diagrams.
As stated above I am a physicist by training, so I apologize in advance for any abuse of language or notation. Any help with these questions would be much appreciated.
For two hooks $\lambda = (p,1^q)$ and $\mu = (r,1^s)$ the skew Schur polynomial $s_{\lambda/\mu}$ is equal to $e_{q-s}h_{p-r}$, as long as $\mu$ is nonempty. It simple to describe in terms of partitions the resulting expression in terms of Schur polynomials (using the Pieri formula), and all of the coefficients in the expansion are $1$. All partitions involved are hooks.
To see that this holds, note that the skew diagram consists of a column of length $q-s$ and a row of length $p-r$ that are edge disjoint. Using the formula for skew Schur polynomials in terms of semistandard tableaux, a tableau on the column is independent of a tableau on the row, so we end up with the product of the sum over all tableaux on the column with the sum over all tableaux on the row. For the column, we get the elementary symmetric function $e_{q-s}$ and for the row we get the complete symmetric function $h_{p-r}$, so the skew Schur polynomial is their product.
For your second question, given the expression above it is clear that we have equality whenever $q-s=q'-s'$ and $p-r=p'-r'$.