I'm collecting different apparently unrelated ways in which the ring (or rather Hopf algebra with $\langle,\rangle$) of symmetric functions $Z[e_1,e_2,\ldots]$ turns up (for a Lie groups course I will be giving next year). So far I have:
*The ring of symmetric functions
*Irreducible representations of symmetric groups =Schur functions
*Irreducible representations of general linear groups = Schur functions
*The homology of $BU$, the classifying space of the infinite unitary group. (It also turns up in several other related generalized homology rings of spectra.)
*The universal commutative $\lambda$-ring on one generator $e_1$
*The coordinate ring of the group scheme of power series $1+e_1x+e_2x^2+\cdots$ under multiplication
What other examples have I missed?