I have a collection of orthogonal polynomials in infinitely commuting variables $x_1, x_2, x_3, \ldots$. I think they must be well known (perhaps Schur or Hermite polynomials or some variant thereof), but I haven't succeeded in finding them in the literature in a form that's recognizable to me. If anyone can point me to an appropriate reference I would be grateful. I suspect the answer to this must be very familiar to many people, but I'm not one of those people.

The polynomials are indexed by Young diagrams (partitions) of all sizes (i.e. [], [1], [2], [1,1], [3], [2,1], [1,1,1], [4], ...).

The measure respect to which they are orthogonal is $$ \prod_{k=1}^\infty \frac{1}{\sqrt{2\pi k}}e^{-\frac{x_k^2}{2k}}dx_k $$ In other words, a product of gaussian measures, with the width proportional to $\sqrt{k}$. (From some points of view it is more natural to replace $x_k$ with $x_k-1$ when $k$ is even; i.e. shift the gaussian to be centered at 1 instead of 0 when $k$ is even.)

The multiplication rule for the polynomials is more complicated than Littlewood-Richardson. Multiplying polynomials corresponding to Young diagrams of sizes $a$ and $b$ results in Young diagrams of sizes ranging from $|a-b|$ to $a+b$. (The highest order part of the multiplication rule is Littlewood-Richardson.) For example $[1] * [2,1] = [2] + [1,1] + [3,1] + [2,2] + [2,1,1]$.

ADDED:

Empirically, it seems to be true that if you sum the polynomials for all Young diagrams of size $n$, weighted by the dimension of the Young diagram, you get the $n$-th Hermite polynomial in the variable $x_1$. (Hat-tip to Suvrit for suggesting that I look at Hermite polynomials.)