# representation theoretic interpretation of Jack polynomials

Monomial symmetric polynomials on $n$ variables $x_1, \ldots x_n$ form a natural basis of the space $\mathcal{S}_n$ of symmetric polynomials on $n$ variables and are defined by additive symmetrization of the function $x^{\lambda} = x_1^{\lambda_1} x_2^{\lambda_2} \ldots x_n^{\lambda_n}$. Here $\lambda$ is a sequence of $n$ nonnegative numbers, arranged in non-increasing order, hence can also be viewed as partition of some integer with number of parts $l(\lambda) \le n$.

Power sum polynomials $p_\lambda$ on $n$ variables also form a basis for $\mathcal{S}_n$

and are defined as $p_\lambda = \prod_{i=1}^n p_{\lambda_i}$, where $p_r = \sum_{i=1}^n x_i^r$.

Schur functions $s_\lambda$ (polynomials) form a basis of the space of symmetric polynomials, indexed by partitions $\lambda$ of at most $n$ parts, and are characterized uniquely by two properties:

1. $\langle s_\lambda, s_\mu \rangle = 0$ when $\lambda \neq \mu$, where the inner product is defined on the power sum basis by $\langle p_\lambda, p_\mu \rangle = \delta_{\lambda,\mu} z_\lambda$, and $z_\lambda = \prod_{i=1}^n i^{\alpha_i} \alpha_i !$, where $\alpha_i$ is the number of parts in $\lambda$ whose lengths all equal $i$. Notice $n!/z_\lambda$ is the size of the conjugacy class in the symmetric group $S_{\sum \lambda_i}$ whose cycle structure is given precisely by $\lambda$.

2. If one writes $s_\lambda$ as linear combination of $m_\mu$'s, then the $m_\lambda$ coefficient is $1$ and $m_\mu$ coefficients are all $0$ if $\mu > \lambda$, meaning the partial sums inequality $\sum_{i=1}^k \mu_i \ge \sum_{i=1}^k \lambda_i$ hold for all $k$ and is strict for at least one $k$. Thus one can say the transition matrix from Schur to monomial polynomial basis is upper triangular with $1$'s on the diagonal.

Jack polynomials generalize Schur polynomials in the theory of symmetric functions by replacing the inner product in the first characterizing condition above with $\langle p_\lambda, p_\mu \rangle = \delta_{\lambda, \mu} \alpha^{l(\lambda)} z_{\lambda}$. The second condition remains the same. It can be thought of as an exponential tilting of the Schur polynomials, and in fact it is intimately connected with the Ewens sampling distribution with parameter $\alpha^{-1}$, a 1-parameter probability measure on $S_n$ or on the set of partitions of $n$ that generalize the uniform measure and the induced measure on partitions respectively.

It turns out that the theory of Schur polynomials has connections with classical representation theory of the symmetric group $S_n$. For instance the irreducible characters of $S_n$ are related to the change of basis coefficient from Schur polynomials to power sum polynomials in the following way:

if we write $s_\lambda = \sum_{\mu} c_{\lambda,\mu} p_\mu$, then $$\chi_\lambda(\mu) = c_{\lambda,\mu} z_\lambda^{-1.}$$.

These are eigenfunctions of the so-called random transposition walk on $S_n$, when viewed as a walk on the space of partitions. The eigenfunctions of the actual random transposition walk on $S_n$ are proportional to the diagonal elements of $\rho$, $\rho$ ranges over all irreducible representations of $S_n$.

The characters $\chi_\lambda$ admit natural generalization in the Jack polynomial setting: simply take the transition coefficients from the Jack polynomials to the poewr sum polynomials. And these when properly normalized indeed gives the eigenfunctions for the so-called metropolized random transposition walk that converges to the Ewens sampling distribution, which is an exponentially tilted 1-parameter family of uniform measure on $S_n$.

My question is, what is the analogue of the diagonal enties of the representations of $\rho$ in the Jack case? Certainly they will be functions on $S_n$.

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I cannot comment on the connection with random walks, but Jack polynomials and, more generally, MacDonald polynomials, arise in representation theory of degenerate affine Hecke algebra, affine Hecke algebra, and double affine Hecke algebra. –  Victor Protsak Aug 27 '10 at 0:10
I have heard of such connections, though I know very little about how it works. What I am interested in is a natural family of $n!$ functions, $f_{\lambda, j}$, $j = 1, \ldots, d_\lambda$, such that $\sum_j f_{\lambda, j}(g) = c_{\lambda, \mu}$ where $\mu$ is the cycle structure of $g$, and when the $\alpha$ parameter becomes $1$, the $f_{\lambda,j}$'s become the diagonal elements of irreducible representations of $S_n$. –  John Jiang Aug 27 '10 at 5:00
$c_{\lambda, \mu}$ of my previous comment post is defined as the change of basis coefficients from Jack to power sum polynomials, i.e., $$J_\lambda = \sum_\mu c_{\lambda,\mu} p_\mu$$. –  John Jiang Aug 27 '10 at 5:01
I also become puzzled by this question, so I allowed myself empty edit for pop it up. –  Alexander Chervov Apr 4 '12 at 12:35
Are Jack polynoms eigenfunctions of quantum calogero-Moser commuting operators ? –  Alexander Chervov Apr 4 '12 at 12:38