TITLE: Avatars of the ring of symmetric polynomials
QUESTION [25 upvotes]: 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?

REPLY [11 votes]: Boson-Fermion correspondence in representation theory
of Kac-Moody algebras (this is implicit in Richard's comments).
Closely related to this: coordinates on the "big cell" in
the loop Grassmannian and it's relation to the KP and KdV
equations where Schur polynomials appear as $\tau$-functions
(see Segal-Wilson old paper in Publ. IHES).