# Canonical bases for modules over the ring of symmetric polynomials

The ring $S=\mathbb{C}[x_1,x_2,\dots,x_n]^{S_n}$ of symmetric polynomials has a number of commonly used bases, but the undisputed world champion of these is the basis consisting of Schur polynomials $s_\lambda$, where $\lambda$ ranges over non-increasing sequences $\lambda_1 \geq \lambda_2 \geq \cdots \geq \lambda_n \geq 0$ of non-negative integers. For a partition $\mu$ of $n$, let $V_\mu$ be the corresponding irreducible $S_n$-module, and let $M(\mu)=(\mathbb{C}[x_1,x_2,\dots,x_n] \otimes V_\mu)^{S_n}$ be the ($S$-module of) $S_n$-invariant polynomial functions on $\mathbb{C}^n$ with values in $V_\mu$. Is there a $\mathbb{C}$-basis of $M(\mu)$ that deserves top billing?

(A bit of background: the dimensions of the homogeneous components of $M(\mu)$ can be computed from the exponents of $V_\mu$, that is, the degrees in which it appears in the coinvariant algebra. There are combinatorial expressions known for these numbers---see e.g. Stembridge's paper "On the eigenvalues of representations of reflection groups and wreath products", Pacific J. Math. 140 (1989), 353--396 and the references therein, but they are not obtained by writing down a particularly nice basis.)

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In my paper Cyclage, catabolism, and the affine Hecke algebra http://arxiv.org/abs/1001.1569 I exhibit a canonical basis for $\mathbb{C}[x_1,x_2,\dots,x_n]$ and more generally a canonical basis for $\mathbb{C}[x_1,x_2,\dots,x_n] \otimes V_\mu$ coming from the extended affine Hecke algebra of type A. The subset of this canonical basis corresponding to cells of shape $(n)$ is a basis for the $S_n$-invariants in this module. For the special case $M(n)$, these canonical basis elements do correspond to Schur functions -- they are Schur functions in the Bernstein generators times $e^+$, where $e^+$ spans the trivial representation for the finite Hecke algebra (see Theorem 6.1).

I have not thought too much about the combinatorics of the canonical basis for $M(\mu)$, but it may be possible to work this out explicitly, including an explicit description of the $S_n$ invariants (see Example 9.21 for a little bit about this). This paper is long and may take some time to get through. Feel free to contact me at the email address at the very bottom of the paper if you have any questions or want to discuss this in detail.

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Thanks! I'll have a look. –  GS Apr 7 '10 at 10:54

One profitable thing to look at might be geometric Satake:

Roughly, one can categorify the symmetric polynomials acting on all polynomials as perverse sheaves on $GL(n,\mathbb{C}[[t]])\setminus GL(n,\mathbb{C}((t)))/GL(n,\mathbb{C}[[t]])$ acting on perverse sheaves on $GL(n,\mathbb{C}[[t]])\setminus GL(n,\mathbb{C}((t)))/I$ where $I$ is the Iwahori (matrices in $GL(n,\mathbb{C}[[t]])$ which are upper-triangular mod $t$).

The maps to polynomials are take a sheaf and send it to the sum over sequences $\mathbf{a}$ of $n$ integers of the Euler characteristic of its stalk at the diagonal matrix $t^{\mathbf{a}}$ times the monomial $x^{\mathbf{a}}$.

One nice thing that happens in this picture is the filtration of polynomials by the invariants of Young subgroups appears as a filtration of categories. Thus, one can take quotient categories and get a nice basis, with lots of good positivity, for the isotypic components.

For the multiplicity space, one might be able to do some trick using cells. It's not immediately clear to me how.

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Thanks; I like geometric Satake and hadn't thought of using it! I'm upvoting, but I hope you don't mind if I don't accept the answer in the (likely vain) hope that someone will come along with a totally different answer. –  GS Jan 21 '10 at 12:09
It's not even a complete answer, so I certanly wouldn't expect you to accept it. –  Ben Webster Jan 21 '10 at 14:15