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 KnopāSahi formula.
A non-symmetric version is also given in [NSS21].

## References

- [NSS21] Yusra Naqvi, Siddhartha Sahi and Emily Sergel. Interpolation polynomials, bar monomials, and their positivity. arXiv e-prints, 2021.
- [Sah94] Siddhartha Sahi. The spectrum of certain invariant differential operators associated to a Hermitian symmetric space. In Lie Theory and Geometry. Birkh\"auser Boston, 1994.