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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.)

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@Kevin: for the curious onlookers like me, could you provide some additional references / context that could aim me in self-edification regarding the background material for this question. (also, are your polynomials some kind of Hermite polynomials?) –  Suvrit Mar 6 '12 at 1:25
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Thanks for the tip on Hermite polynomials. These seem to be some sort of higher dimensional variant. As for references or context for self-edification... I'm not really an expert on orthogonal polynomials or representation theory, which is why I'm asking a potentially elementary question here. From my point of view the context is TQFTs and skein modules. The above polynomials bear the same relation to the Birman-Wenzl-Murikami category as Chebyshev polynomials bear to the Temperley-Leb category. A good starting point for this subject is... –  Kevin Walker Mar 6 '12 at 2:52
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...the book "Temperley-Lieb recoupling theory and invariants of 3-manfolds" by Kauffman and Lins. After that one might want to look at various papers of Hugh Morton from the 1990s (or 2000s?). But that's my own peculiar point of view on the subject, not the standard one. –  Kevin Walker Mar 6 '12 at 2:55
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Have you tried weighting by a different column of the character table of the symmetric group? –  John Wiltshire-Gordon Mar 6 '12 at 8:04
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There is a good chance that they are special cases of Macdonald polynomials of type $A_n$. Check Macdonald's book on symmetric functions and Hall polynomials. –  Richard Borcherds Mar 6 '12 at 21:47

1 Answer 1

The answer appears to be as follows. ("Appears" because I haven't yet written out a detailed proof.)

Let $H^{[k]}_n(x)$ denote the variant of Hermite polynomials which are orthogonal with respect to the measure $$ \frac{1}{\sqrt{2\pi k}}e^{-x^2/2k}dx . $$ Since the measure in the question is a product of the above measures (over all positive integers $k$), we have a family of orthogonal multivariable polynomials $$ H^{[1]}_{n_1}(x_1) H^{[2]}_{n_2}(x_2) \cdots H^{[j]}_{n_j}(x_j) ,$$ indexed by tuples $(n_1,\ldots,n_l)$. The orthogonal polynomials of the questions are linear combinations of these. More specifically, let $N = \sum_i i\cdot n_i$. Think of $(n_1,\ldots,n_l)$ as encoding a conjugacy class in the symmetric group $S_N$, where $n_i$ is the number of $i$-cycles in a permutation. We can use the character table of $S_N$ to change basis from conjugacy-class-bump-functions to characters-of-representations. Applying an analogous change of basis to the above products of Hermite polynomials (separately for each $N$) yields the polynomials described in the question. There are some normalization factors I have not mentioned, related to the fact that some of the bases mentioned above are orthogonal but not orthonormal.

Thanks again to Suvrit and John Wiltshire-Gordon for pointing me in the right direction. I have not yet looked into Richard Borcherds' suggestion that these might also be specializations of Macdonald polynomials.

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