One interesting fact in symmetric function theory is that the power symmetric function $p_n$ can be written as an alternating sum of hook Schur functions $s_{\lambda}$: $$ p_n = \sum_{k+\ell = n} (-1)^\ell s_{k, 1^\ell}. $$

A priori, all that is known is that $p_n$ can be expressed as a sum of Schur functions $s_{\lambda}$, with the sum ranging over all partitions of $n$, not just those of hook shape. The fact that the coefficients of all non-hook shapes are zero is quite interesting and most likely says something about representations of $S_n$.

However, this property does not extend to other generalizations of the Schur functions. In particular, when $p_n$ is expanded in the Jack basis $J_{\lambda}$ of symmetric functions with coefficients in $\mathbb{Q}(\alpha)$, such a vanishing phenomenon no longer occurs. In particular, the coefficient of $J_{2,2}$ in the expansion of $p_4$ is non-zero (though, of course, it vanishes when $\alpha = 1$).

Is there a representation-theoretic or geometric explanation for why the expansion of power functions in the Schur basis has such a nice form? And is there any sort of generalization that could determine which polynomials in the $p_n$ would have a similar property in one of the generalized bases, specifically the Jack basis? Though it may be too much to ask for, I would love to have a combinatorially explicit, algebraically independent set of polynomials in the $p_i$ that generate the ring of symmetric functions in $\mathbb{Q}(\alpha)$ with the property that each generator is a sum of hook Jack symmetric functions only.