# The symmetric functions catalog

An overview of symmetric functions and related topics

2021-04-20

## Jack interpolation polynomials

The Jack interpolation polynomials $\jackP^{\rho}_\lambda(\xvec)$ are non-homogeneous extensions of the Jack polynomials, where $\rho = (\rho_1,\rho_2,\dotsc)$ is a sequence of indeterminates. These polynomials were first introduced by S. Sahi in 1994, see [Sah94]. Similar to the shifted Jack polynomials, the Jack interpolation polynomials are characterized by relatively simple vanishing conditions.

### Definition

The polynomial $\jackP^{\rho}_\lambda(x_1,\dotsc,x_n),$ is the unique symmetric function of total degree $|\lambda|$ which satisfies the following.

• For every integer partitions $\mu,$ $|\mu| \leq |\lambda|$ and $\mu \neq \lambda,$ we have $\jackP^{\rho}_\lambda(\mu + \rho) = 0.$
• The coefficient of $\monomial_\lambda$ in $\jackP^{\rho}_\lambda$ is $1.$

Let $\delta = (n-1,n-2,\dotsc,1,0).$ In the case $\rho = r \delta,$ one have that

$\jackP^{r \delta}_\lambda(\xvec) = \jackP_\lambda(\xvec; 1/r) + \text{ lower order terms}.$

That is, a particular choice of $\rho$ allows us to recover the classical Jack $P$ polynomial $\jackP_\lambda(\xvec; 1/r),$ as the highest term.

### Monomial positivity

In [NSS21], the authors study the normalized version

$\jackJ^{r \delta}_\lambda(\xvec) \coloneqq H_\lambda(a) \cdot (-1)^{|\lambda|} \jackP^{r \delta}_\lambda(-\xvec),$

where $H_\lambda$ is an $a$-deformation of hook products, and $r = 1/a.$

They show that the coefficients $c_{\mu}(a)$ in the expansion

$\jackJ^{r \delta}_\lambda(\xvec) = \sum_{\mu} a^{|\mu|-|\lambda|} c_{\mu}(a) \monomial_\mu(\xvec)$

are elements in $\setN[a].$

In fact, in [Thm. 5.4, NSS21] the authors give a combinatorial expansion,

$\jackJ^{r \delta}_{\gamma^+}(\xvec) = \sum_{\text{T admissible}} d_T(a) \xvec^{\underline{w(T)}},$

where $\xvec^{\underline{w(T)}}$ is a certain bar monomial, which is monomial positive. This formula is the interpolation analog of the Knop–Sahi formula. A non-symmetric version is also given in [NSS21].