The plethysm $s\_{\nu}[s\_{\mu}]$ of two symmetric functions is the character of the composition of Schur functors $S^{\nu}(S^{\mu}(V))$. We know that this operation is linear and multiplicative in its first argument. But is there a way to develop


1. $s\_{\nu}[s\_{\mu} + s\_{\lambda}]$;
2. $s\_{\nu}[s\_{\mu}s\_{\lambda}]$;


in terms of plethysms $s\_{\nu}[s\_{\mu}]$ and $s\_{\nu}[s\_{\lambda}]$ ? I think I have heard of a formula for the first one, but I don't find it anymore!



Interaction of plethysm with other operations
In principle one can develop (1) using the coproduct in the ring of symmetric functions. By the Littlewood–Richardson rule, $\Delta(s\_\nu) = \sum\_{\alpha}\sum\_\beta c^\nu\_{\alpha\beta} s\_\alpha \otimes s\_\beta$ where $c^\nu\_{\alpha\beta}$ is a Littlewood–Richardson coefficient, and correspondingly


$$s\_\nu[s\_\lambda + s\_\mu] = \sum\_{\alpha}\sum\_\beta c^\nu\_{\alpha\beta} s\_\alpha[s\_\lambda] s\_\beta[s\_\mu].$$


Here the sum is over all partitions such that $|\alpha|+|\beta| = |\nu|$.
Somewhat similarly,


$$s\_\nu[s\_\lambda s\_\mu] = \sum\_{\alpha}\sum\_{\beta} k^\nu\_{\alpha\beta} s\_\alpha[s\_\lambda] s\_\beta[s\_\mu]$$


where the sum is over all partitions $\alpha$ and $\beta$ of $|\nu|$ and $k^\nu\_{\alpha\beta}$ is the Kronecker coefficient, most easily defined as the inner product $\langle \chi^\nu, \chi^\alpha \chi^\beta \rangle$ in the character ring of the symmetric group. Equivalently the$k^\nu\_{\alpha\beta}$ are the structure constants for the internal product, usually denoted $\star$, on the ring of symmetric functions. These formulae can be found in MacDonald's textbook: see (8.8) and (8.9) on page 136, and hold replacing $s\_\lambda$ and $s\_\mu$ with arbitrary symmetric functions.


In practice, at least in my experience, this usually leads to a mess. One special case that's worth noting is when $\nu = (n)$, in which case the Littlewood—Richardson coefficient is non-zero only if $\alpha = (m)$ and $\beta = (n-m)$ for some $m \in \{0,1,\ldots, n\}$ and we get


$$s\_{(n)}[s\_\lambda + s\_\mu] = \sum\_m s\_{(m)}[s\_\lambda] s\_{(n-m)}[s\_\mu].$$


This is the symmetric function version of $\mathrm{Sym}^n (V \oplus W) = \sum\_{m=0}^n \mathrm{Sym}^m V \otimes \mathrm{Sym}^{n-m} W$ for polynomial representations of $\mathrm{GL}\_d(\mathbb{C})$. There is a corresponding rule for exterior powers and so for $s\_{(1^n)}$.


This also gives one indication that (2) is even harder: [one related question](https://mathoverflow.net/q/321052/7709) was asked on MathOverflow. Example 3 on page 137 of MacDonald gives the special case for $\nu = (n)$, when $\chi^{(n)}$ is the trivial character, and so $\langle \chi^{(n)}, \chi^{\alpha}\chi^{\beta}\rangle = \langle \chi^{\alpha}, \chi^\beta\rangle = [\alpha=\beta]$. Hence


$$s\_{(n)}[s\_\lambda s\_\mu] = \sum\_{\alpha} s\_\alpha[s\_\lambda] s\_\alpha[s\_\mu]. $$


Great care is needed when extending these rules to arbitrary symmetric functions. For instance, $s\_\nu[-f] = (-1)^{|\nu|} s\_{\nu'}[f]$ for any symmetric function $f$ and, as Richard Stanley points out in a comment below, the expression $s\_\nu[f-f]$ should be interpreted as a plethystic substitution using the alphabets for $f$ and $-f$, not as $s\_\nu[0]$; correctly interpreted, it can be expanded using the coproduct rule and the rule for $s\_\nu[-f]$ just given.


