co.combinatorics - Canonical bases for modules over the ring of symmetric polynomials

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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