@QiaochuYuan Sorry, my mistake (now corrected). From the first equation you set $x=y=1$ in order to find the second equation (which holds whether you're using the sum or multiplicative group laws), which gives you that only one such $s_{i,j}$ is non-zero (and thus equal to unity and must be such $i=j=\alpha$). Then returning to general $x$ and $y$ the only remaining question is which $\alpha$ are possible.

@Jlamprong This is not a place to get your homework done for you, rather research-level mathematics, I just thought I'd help you out. You might also have typed $n\times m$ the wrong way around, or there could be a typo in whereever you got this problem from. Please be willing to put in some work yourself and in future take basic queries that aren't research-level to math.stackexchange.org instead.

@QiaochuYuan in which case the argument runs just as above and leads to the same conclusion, just that this time we find $q(x)q(y)=q((xy)^\alpha)$ rather than the same case with $f$.

@Jlamprong Since $K$ is full rank and has $m\leq n$ then it has a left inverse, as explained here: math.stackexchange.com/questions/108612/…. Since $K^T$ is also of full rank, but with rows and columns swapped it has a right inverse. I realise that these are the wrong inverses, we want the left inverse of $K^T$ and the right inverse of $K$. Are you sure you didn't mean $m>n$?

This sounds very promising, thank you. Unfortunately it rather stretches my understanding beyond breaking point. I am sure I can read up on formal group laws (thank you for the reference), but I don't see where the link is to monoidal structures over $\mathbf{Set}$. In particular, googling "analytic bifunctor" (with quotes) only returns this page.

@StephanMüller Hi. I did see that, and I linked to that thread in my question. Since I'm interested in the answer I thought I'd see if any of the MO community knew of any results (maybe ones since August 2008). Incidentally, I don't understand how that the paper that Stephen Lack mentions on that thread (sciencedirect.com/science/article/pii/0022404980900821) is linked to $\mathbf{Set}$.