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#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 polynomials 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 [Eq. (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 [KS96] that \(\jackShifted_\mu\) is the unique polynomial in \(\Lambda^{a}\) of degree at most \(|\mu|\) such that \[\jackShifted_\mu(\lambda) = \begin{cases} a^{-|\mu|} H'_\mu, &\lambda = \mu \\ 0, \quad &|\mu| \geq |\lambda|, \; \mu \neq \lambda. \end{cases}\] (Note that the inequality is in the wrong direction in [p. 70, OO97].)

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

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 formula

Just as 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]. \[\jackShifted_\mu(\xvec;a) = \sum_{T \in \mathrm{RSSYT}(\mu)} \psi_T(a) \prod_{s \in \mu} (x_{T(s)} - \arm'(s) + \leg'(s)/a),\] 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 \(a = 1,\) \(c^{\lambda}_{\mu\nu}(1)\) coincides with the usual Littlewood–Richardson coefficient.

From the vanishing result by F. Knop and S. Sahi [KS96] and a simple contradiction argument, see [p. 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 [AF19].

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 \[\psi'_T(\alpha) \coloneqq \prod_{i=1}^n \psi'_{\rho^i/\rho^{i-1}}(\alpha)\] 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 \[c^{\lambda}_{\mu\nu} = \frac{1}{|\lambda|-|\nu|}\left( \sum_{\nu \to \nu^+} \psi'_{\nu^+ / \nu} c^{\lambda}_{\mu \nu^+} - \sum_{\lambda^- \to \lambda } \psi'_{\lambda / \lambda^-} c^{\lambda^-}_{\mu \nu} \right)\] 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 (1.4 MB).

For example, in the data one can find that \(\mathtt{JackStructureConstant}_{311,222,42211}(a)\) is equal to \[\begin{aligned} & 192a^4(1 + a)^2(3 + a)(4 + a)(1 + 2a)(2 + 3a) \cdot \\ & (45 + 276a + 547a^2 + 426a^3 + 134a^4 + 12a^5). \end{aligned}\] Note that some of these constants are Laurent polynomials in \(a.\)

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