Avatars of the ring of symmetric polynomials - MathOverflow

Avatars of the ring of symmetric polynomials

2011-04-29T16:10:59Z

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?

Answer by David Speyer for Avatars of the ring of symmetric polynomials

2011-04-29T17:05:59Z

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.

Answer by S. Carnahan for Avatars of the ring of symmetric polynomials

2011-04-29T17:19:43Z

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.

Answer by Qiaochu Yuan for Avatars of the ring of symmetric polynomials

2011-04-29T18:09:13Z

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.

Answer by Richard Stanley for Avatars of the ring of symmetric polynomials

2011-04-30T00:40:12Z

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

Answer by James Borger for Avatars of the ring of symmetric polynomials

2011-04-30T00:43:19Z

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.

Answer by Maarten Bergvelt for Avatars of the ring of symmetric polynomials

2011-04-30T01:22:25Z

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.

Answer by Allen Knutson for Avatars of the ring of symmetric polynomials

2011-04-30T02:47:08Z

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?)

Answer by Victor Ginzburg for Avatars of the ring of symmetric polynomials

2011-04-30T04:31:30Z

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).