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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?

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You do know of Hazewinkel's texts? – darij grinberg Apr 29 '11 at 16:18
Also I have tried, some time ago, to show or even formulate the conjecture that your ring is the ring of the endofunctors of the category of all finite sets, suitably extended (the way $\mathbb N$ is extended to obtain $\mathbb Z$) and factored by some kind of canonical isomorphisms. The idea is that multiplication is cartesian product, addition is direct sum, $\lambda^i$ is taking the set of all $i$-element subsets. Unfortunately, I do not really understand what we should factor out. – darij grinberg Apr 29 '11 at 16:26
Schur proved something like this, except using finite dimensional vector spaces rather than finite sets. – Richard Borcherds Apr 29 '11 at 17:55
Oh right, I see this is what Qiaochu posted. – darij grinberg Apr 29 '11 at 18:15

The cohomology ring of the Grassmannian, an avatar known to Philip Hall.

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Grassmannians are more or less the same as BU, though this is not trivial so I guess it counts as another independent example. I'm slightly surprised Hall knew this as I've seen no sign in his papers that he knew anything about topology. – Richard Borcherds Apr 30 '11 at 12:33
I am referring to Hall's obscure but influential paper "The algebra of partitions," Proceedings of the 4th Canadian Mathematical Congress, Banff, 1959, pp. 147–159. – Richard Stanley Apr 30 '11 at 18:54

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).

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As the nLab says, it's the Grothendieck ring of the category of $k$-linear species (functors $\text{FinSet}_0 \to k\text{-Vect}$) for $k$ a field of characteristic zero. See also Schur functor.

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Any $\lambda$-ring has a commuting family of Frobenius lifts (the Adams operators) indexed by the primes. Conversely, by Wilkerson's theorem, any torsion-free ring with a commuting family of Frobenius lifts admits a unique $\lambda$-ring structure. So the ring $\Lambda$ of symmetric functions can also be viewed as the free object on one generator in the category of torsion-free rings with commuting Frobenius lifts.

Also, $\Lambda$ represents the big Witt vector functor. In other words, it's the coordinate ring of the big Witt ring scheme.

This point of view is more arithmetic than the usual approach to symmetric functions, and in my opinion the connection between the two is still a bit mysterious. For instance, the definition of $\lambda$-ring above can obviously be generalized to any global field (and hence so can the big Witt functor) using $p^f$-power Frobenius lifts modulo prime ideals, but I don't know whether reasonable analogues of the elementary symmetric functions exist in these generalizations.

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The Hall algebra of finite abelian $p$-groups, "at $p=1$". Wikipedia explains the basics here, lecture 2 of Schiffmann's lectures on Hall algebras has more.

A more high-tech way to thinking of setting $p=1$ is to consider the Hall algebra of nilpotent $\mathbb{C}[t]$ modules (nilpotent meaning that $t$ acts nilpotently), where, instead of counting subgroups, you take the Euler characteristic of the space of subgroups.

UPDATE: I have realized that I don't understand this as well as I thought. I know that the symmetric functions are lurking here, but I don't quite understand how. You are probably better reading Schiffmann than trusting me for details.

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A good reference maybe? – darij grinberg Apr 29 '11 at 17:38
Macdonald's book is a good reference for these. If I remember correctly you need to work with the symmetric functions over polynomials in q or something like that, and if you set q=1 you recover the usual symmetric functions. – Richard Borcherds Apr 29 '11 at 17:49
Lecture 2 from Schiffman's lectures on Hall algebras is pretty good. – David Speyer Apr 29 '11 at 18:10
Thanks, these look really nice. – darij grinberg Apr 29 '11 at 18:16

You can make a free bosonic vertex algebra out of this ring. If you base change to the complex numbers, the resulting object should describe some aspect of a closed bosonic string propagating on a real line. If I'm not mistaken, the states are symmetric functions in harmonic oscillator modes.

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There is no need to work over the complex numbers: the integral ring of symmetric functions gives an integral form of the vertex algebra. Though the harmonic oscillator modes do not generate this integral form. – Richard Borcherds Apr 29 '11 at 17:53
Sorry, I meant that the vertex algebra was defined over the integers, but that the connection to string theory involves base change. – S. Carnahan Apr 29 '11 at 17:59

Aguiar and Marajan, in their book Monoidal Functors, Species and Hopf Algebras, obtain the Hopf algebra of symmetric functions (and many other Hopf algebras) by applying a "Fock Functor" to a Hopf monoid in the category of species (functors from finite sets and bijections to vector spaces over a field). This seems different from but related to the construction mentioned by Qiaochu Yuan.

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The direct sum over $n$ of the total homology of the Hilbert scheme of $n$ points in the plane. (Reference: Nakajima's book.)

The (stably-almost-)complex cobordism ring of a point. I expect that's pretty close to the power series thing you mentioned, via the connection to the universal formal group law. (I wonder if there are nice manifolds corresponding to the Schur functions?)

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Is there a natural way to identify $MU^*$ with such a ring of invariants? I'm not aware of one. – Dev Sinha Apr 30 '11 at 4:10
Bother, I meant to ask you! Can you identify it with any of the other avatars here, like $H^\ast(BU)$? – Allen Knutson May 2 '11 at 21:05
The cohomology operations $MU^*(MU)$ are related to $MU^*(BU)$ by a Thom isomorphism and the Atiyah-Hirzebruch spectral sequence from $H^*(BU;MU^*)$ to $MU^*MU$ collapses. For $MU^*(pt)\otimes \mathbb{Q}$ there is a pairing with polynomials defined using symmetric functions. – Ben Cooper Apr 6 '13 at 13:19

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