The symmetric functions catalog

#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 has 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. The same paper proves the non-symmetric interpolation version of this positivity statement. In that setting partitions are replaced by compositions, and the positive bar-monomial expansion is governed by the bar order and by glissades.

#Binomial and Littlewood–Richardson coefficients

H. Chen and S. Sahi study binomial coefficients and Littlewood–Richardson coefficients for interpolation polynomials [CS24]. Their results include positivity and monotonicity for binomial coefficients, partial positivity for interpolation Littlewood–Richardson coefficients, and weighted sum formulas for both kinds of coefficients. As an application, they prove a symmetric-function inequality for the containment order on partitions in the shifted normalized Jack basis.

Bibliography

1. [CS24]Hong Chen and Siddhartha Sahi. Interpolation Polynomials, Binomial Coefficients, and Symmetric Function Inequalities. arXiv:2403.02490, 2024.
   .bib

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   @article{ChenSahi2024x,
     author = {Hong Chen and Siddhartha Sahi},
     title = {Interpolation {P}olynomials, {B}inomial {C}oefficients, and
       {S}ymmetric {F}unction {I}nequalities},
     year = {2024},
     eprint = {2403.02490},
     url = {https://arxiv.org/abs/2403.02490},
     journal = {arXiv e-prints}
   }
   

2. [NSS21]Yusra Naqvi, Siddhartha Sahi and Emily Sergel. Interpolation polynomials, bar monomials, and their positivity. arXiv:2104.08598, 2021.
   .bib

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   @article{NaqviSahiSergel2021x,
   Author = {Yusra Naqvi and Siddhartha Sahi and Emily Sergel},
   Title = {Interpolation polynomials, bar monomials, and their positivity},
   Year = {2021},
   Eprint = {2104.08598},
     url = {https://arxiv.org/abs/2104.08598},
   journal = {arXiv e-prints}
   }
   

3. [Sah94]Siddhartha Sahi. The spectrum of certain invariant differential operators associated to a Hermitian symmetric space. Lie theory and geometry:569–576, 1994.
   .bib

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   @incollection{Sahi1994,
     doi = {10.1007/978-1-4612-0261-5_21},
     url2 = {https://doi.org/10.1007/978-1-4612-0261-5_21},
     year = {1994},
     publisher = {Birkh\"{a}user Boston},
     pages = {569--576},
     author = {Siddhartha Sahi},
     title = {The Spectrum of Certain Invariant Differential Operators Associated to a {H}ermitian Symmetric Space},
     booktitle = {Lie Theory and Geometry}
   }