2024-02-22

## Shifted Jack polynomials

The shifted Jack polynomials generalize the Jack polynomials, as well as the shifted Schur polynomials. The shifted Jack polynomials constitute a basis for the ring of shifted symmetric functions $\Lambda^{a}.$ This is the ring of polyomials symmetric in variables $x_i - i/a.$ Note that these functions are not homogeneous.

They are given as a limit of interpolation Macdonald P polynomials.

### Definition

Define two $a$-deformations of the hook products, $H_\lambda$ and $H'_\lambda,$ as

\[ H_\lambda = \prod_{s \in \lambda} (a \cdot \arm_\lambda(s) + \leg_\lambda(s) + 1), \quad H'_\lambda = \prod_{s \in \lambda} (a \cdot \arm_\lambda(s) + \leg_\lambda(s) + a). \]These are denoted $c_\lambda(a)$ and $c'_\lambda(a)$ in [(10.22), Mac95].

The polynomials $\jackShifted_\mu$ lie in $\Lambda^{a}(x_1,\dotsc,x_n)$ and it was shown by F. Knop and S. Sahi in [KS96] that $\jackShifted_\mu$ is the unique polynomial in $\Lambda^{a}$ of degree at most $|\mu|$ such that

\begin{equation*} \jackShifted_\mu(\lambda) = \begin{cases} a^{-|\mu|} H'_\mu, &\lambda = \mu \\ 0, \quad &|\mu| \geq |\lambda|, \; \mu \neq \lambda. \end{cases} \end{equation*}(Note that the inquality is in the wrong direction in [p.70, OO97].)

We also define the shifted version of $\jackJ_\mu(\xvec;a)$ as $\jackShiftedJ_\mu(\xvec,a) \coloneqq H_\mu \jackShifted_\mu(\lambda).$

**Example (Expression for $\jackShifted_{(2,1)}(\xvec,a)$).**

The polynomials get rather complicated.

\[ (a^2 x_1 x_2^2+a^2 x_1 x_3^2+a^2 x_2 x_3^2+a^2 x_1^2 x_2-2 a^2 x_1 x_2+a^2 x_1^2 x_3+a^2 x_2^2 x_3-2 a^2 x_1 x_3-2 a^2 x_2 x_3+2 a x_1 x_2^2+a x_2^2+2 a x_1 x_3^2+2 a x_2 x_3^2+2 a x_3^2+2 a x_1^2 x_2-3 a x_1 x_2-2 a x_2+2 a x_1^2 x_3+2 a x_2^2 x_3-5 a x_1 x_3+6 a x_1 x_2 x_3-7 a x_2 x_3-4 a x_3+2 x_2^2+4 x_3^2+2 x_1 x_2-4 x_2+4 x_1 x_3+6 x_2 x_3-14 x_3)/(a(a+2)) \]### RSSYT definition

Similar to how the Jack polynomials can be defined as a sum over reverse semistandard Young tableaux, there is an analogous formula for the shifted Jack polynomials, see [OO97].

\begin{equation*} \jackShifted_\mu(\xvec;a) = \sum_{T \in \mathrm{RSSYT}(\mu)} \psi_T(a) \prod_{s \in \mu} (x_{T(s)} - \arm'(s) + \leg'(s)/a), \end{equation*}where $\arm'(s) = j-1$ and $\leg'(s) = i-1$ for the box $s=(i,j).$ Comparing this with the similar-looking formula for the Jack P polynomials, we see that the top degree component of $\jackShifted_\mu(\xvec;a)$ is given by the ordinary Jack function $\jackP_\mu(\xvec;a).$

See also the corresponding formula for the interpolation Macdonald P polynomials.

## Littlewood–Richardson coefficients

Define the Littlewood–Richardson type structure constants, $c^{\lambda}_{\mu\nu},$ which are rational functions in $a,$ by the relation

\[ \jackShifted_\mu \jackShifted_\nu = \sum_\lambda c^{\lambda}_{\mu\nu}(a) \jackShifted_\lambda. \]If $|\lambda|=|\mu|+|\nu|,$ then $c^{\lambda}_{\mu\nu}(a) = \langle \jackP_\mu \jackP_\nu, \jackP_\lambda \rangle_a,$ the ordinary Jack structure constants. For $a = 1,$ $c^{\lambda}_{\mu\nu}(1)$ coincide with the usual Littlewood–Richardson coefficients.

From the vanishing result by F. Knop and S. Sahi [KS96] and a simple contradiction argument (see [top of page 4434, MS99] for the case $\alpha=1$), it follows that $c^{\lambda}_{\mu\nu}(a)$ is identically zero unless $\lambda \supseteq \mu$ and $\lambda \supseteq \nu.$

### Generalization of Stanley's conjecture

The following section contains information from [AF19a].

Let $g^{\lambda}_{\mu \nu}(a) \coloneqq H'_\lambda H_\mu H_\nu c^{\lambda}_{\mu\nu}(a).$

**Conjecture (Alexandersson, Féray (2014)).**

The expression $a^{|\lambda|-|\mu|-|\nu|-2} g^{\lambda}_{\mu\nu}(a)$ is a polynomial in $\setN[a].$

Note that whenever $|\lambda|=|\mu|+|\nu|,$ this conjecture implies Stanley's conjecture [Sta89] regarding structure constants for Jack polynomials.

**Proposition (P. Alexandersson, V. Féray (2019)).**

The expression $a^{|\mu|+|\nu|-|\lambda|-2} g^{\lambda}_{\mu\nu}$ is a polynomial in $a.$

### Recursion for the $c^{\lambda}_{\mu\nu}(a)$

One can prove that $c^{\lambda}_{\mu\lambda}(a) = \jackShifted_\mu(\lambda,a).$

Define $\psi'_T(\alpha),$ which is similar to $\psi_T(\alpha)$ and calculated as

\begin{equation*} \psi'_T(\alpha) \coloneqq \prod_{i=1}^n \psi'_{\rho^i/\rho^{i-1}}(\alpha) \end{equation*}and where $\psi'_{\lambda/\mu}(\alpha) \coloneqq \psi_{\lambda'/\mu'}(1/\alpha).$

The following proposition can be proved by applying the same technique as in [MS99].

**Proposition (See [Sah11]).**

Let $\mu, \nu \subseteq \lambda.$ Then

\begin{equation*} c^{\lambda}_{\mu\nu} = \frac{1}{|\lambda|-|\nu|}\left( \sum_{\nu \to \nu^+} \psi'_{\nu^+ / \nu} c^{\lambda}_{\mu \nu^+} - \sum_{\lambda^- \to \lambda } \psi'_{\lambda / \lambda^-} c^{\lambda^-}_{\mu \nu} \right) \end{equation*}where the first sum is taken over all possible ways to add one box to the diagram $\nu,$ and the second sum is over all ways to remove one box from $\lambda.$

This together with the identity $c^{\lambda}_{\mu \lambda} = \jackShifted_\mu(\lambda)$ gives a recursive method to compute the $c^{\lambda}_{\mu\nu}.$

### Data for structure constants

Let

\[ \mathtt{JackStructureConstant}_{\mu,\nu,\lambda}(a) \coloneqq H'_\lambda H_\mu H_\nu c^{\lambda}_{\mu\nu}(a) \]be the structure constants $g^{\lambda}_{\mu \nu}(a)$ appearing in the generalized Stanley conjecture. We have computed these for all $\mu,\nu$ with size at most $6,$ and the data is available for download here (1.4Mb).

For example, in the data you can find that $\mathtt{JackStructureConstant}_{311,222,42211}(a)$ is equal to

\begin{align} & 192a^4(1 + a)^2(3 + a)(4 + a)(1 + 2a)(2 + 3a) \cdot \\ & (45 + 276a + 547a^2 + 426a^3 + 134a^4 + 12a^5). \end{align}Note that some of these constants are Laurent polynomials in $a.$

## References

- [AF19a] Per Alexandersson and Valentin Féray. A positivity conjecture on the structure constants of shifted Jack functions. arXiv e-prints, 2019.
- [KS96] Friedrich Knop and Siddhartha Sahi. Difference equations and symmetric polynomials defined by their zeros. International Mathematics Research Notices, 1996(10):473, 1996.
- [Mac95] Ian G. Macdonald. Symmetric functions and Hall polynomials. Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, Second edition, 1995. With contributions by A. Zelevinsky, Oxford Science Publications
- [MS99] Alexander I. Molev and Bruce E. Sagan. A Littlewood–Richardson rule for factorial Schur functions. Transactions of the American Mathematical Society, 351(11):4429–4444, November 1999.
- [OO97] A. Okounkov and G. Olshanski. Shifted Jack polynomials, binomial formula, and applications. Mathematical Research Letters, 4(1):67–78, 1997.
- [Sah11] S. Sahi. Binomial coefficients and Littlewood–Richardson coefficients for Jack polynomials. International Mathematics Research Notices, 2011:1597–-1612, 2011.
- [Sta89] Richard P. Stanley. Some combinatorial properties of Jack symmetric functions. Advances in Mathematics, 77(1):76–115, September 1989.