The symmetric functions catalog

#Jack P polynomials

The Jack polynomials are a family of symmetric functions which extends the Schur polynomials. They were introduced by H. Jack in [Jac70]. They are indexed by integer partitions and constitute a basis for the space of symmetric functions. For an overview, see [Mac95] and [Sta89].

The Jack polynomials can be generalized to the shifted Jack polynomials, Jack interpolation polynomials and Macdonald \(P\) polynomials.

#Deformed Hall inner product

Let \(\langle \cdot, \cdot \rangle_a\) be the inner product on symmetric functions such that \[\langle \powerSum_\lambda, \powerSum_\mu \rangle_a = \delta_{\lambda\mu} a^{\length(\lambda)} z_\lambda.\] Then the family \(\jackP_\lambda(x;a)\) is the unique family that satisfies:

• Orthogonality: \(\langle \jackP_\lambda, \jackP_\mu \rangle_a = 0\) whenever \(\lambda \neq \mu.\)

• Triangularity: \(\jackP_\lambda = \sum_{\mu \lt_d \lambda } c_{\lambda \mu} \monomial_\mu.\)

• Normalization: \([\monomial_{1^n}]\jackP_\lambda = 1.\)

These symmetric functions have coefficients which are rational functions in \(a.\)

#RSSYT formula

The Jack polynomial \(\jackP_\mu(\xvec;a)\) in \(n\) variables may be defined as \[\jackP_\mu(\xvec;a) = \sum_{T \in \textrm{RSSYT}(\mu)} \psi_T(a) \prod_{s \in \mu} x_{T(s)},\] where the sum is taken over all reverse tableaux with entries in \([n]\) and shape \(\mu.\) Here, \(\psi_T(a)\) is the rational function defined in [Mac95] as \[\psi_T(a) \coloneqq \prod_{i=1}^n \psi_{\rho^i/\rho^{i-1}}(a)\] where \(\rho^i/\rho^{i-1}\) is the skew shape formed by the boxes of \(T\) containing \(i\) (\(\rho^0 = \emptyset\)) and \[\psi_{\lambda/\mu}(a) \coloneqq \prod_{s \in R_{\lambda/\mu} \setminus C_{\lambda/\mu} } \frac{ (a \cdot\arm_\lambda(s) + \leg_\lambda(s) + a)(a \cdot \arm_\mu(s) + \leg_\mu(s) + 1) }{ (a\cdot \arm_\lambda(s) + \leg_\lambda(s) + 1)(a \cdot\arm_\mu(s) + \leg_\mu(s) + a) }.\] Here, \(R_{\lambda/\mu}\) denotes the set of boxes in a row that intersects the shape \(\lambda/\mu.\) The set of boxes \(C_{\lambda/\mu}\) is defined in a similar manner for columns.

This formula generalizes to the super Jack polynomials, see [SV05].

H. B. Dali studies the expansion of Jack polynomials in the power-sum basis through decorated bipartite maps [Dal23]. The paper proposes an injectively decorated variant of the Dołęga–Ben Dali map expansion and proves it for Jack polynomials indexed by two-column partitions. G. Chapuy and M. Dołęga deform Schur functions to Jack functions inside the \(2\)-Toda tau-function for weighted Hurwitz numbers [CD22]. They prove that the resulting coefficients are polynomials in the deformation parameter \(b\) with nonnegative integer coefficients, interpreted through generalized branched coverings by orientable and non-orientable surfaces.

N. Wang and K. Wu define 3-Jack polynomials attached to three-dimensional Young diagrams [WW23]. These depend on three axis parameters and generalize Schur and Jack polynomials to a three-dimensional setting; they can be described using vertex operators and Pieri formulas.

#Recursive formula

There is an efficient recursion for computing the Kostka coefficients \(K_{\lambda\mu}(\alpha),\) appearing in the expansion \(\jackP_\lambda = \sum_{\mu} K_{\lambda\mu}(\alpha) \monomial_\mu.\) This recursion can be found in [p. 327, Mac95], where Macdonald uses the notation \(u_{\lambda\mu}\) for these coefficients. See also [LLM99, Rob00], and [Prop. 2.16, DES07], where this formula appears. Generalizations to other root systems can be found in [DLM04].

#Cauchy identity

The (dual) Cauchy identity for Jack \(P\) polynomials states that \[\sum_{\lambda} \jackP_\lambda(x;a) \jackP_\lambda(y;1/a) = \prod_{i,j} (1+x_i y_j),\] see [Eq. (2.6), Mac92]. There is also a generalization of the Cauchy identity for the Jack J polynomials.

#Jack J polynomials

The integral form Jack polynomials are defined as the unique family satisfying the following relations:

• Orthogonality: \(\langle \jackJ_\lambda, \jackJ_\mu \rangle_a = 0\) whenever \(\lambda \neq \mu.\)

• Triangularity: \(\jackJ_\lambda = \sum_{\mu \lt_d \lambda } c_{\lambda \mu} \monomial_\mu.\)

• Normalization: \([\monomial_{1^n}]\jackJ_\lambda = n!.\)

These symmetric functions have coefficients in \(\setN[a]\) in the augmented monomial basis. Equivalently, if \[\jackJ_\lambda(\xvec;a) = \sum_{\mu} v_{\lambda\mu}(a) \monomial_\mu(\xvec) \quad\text{and}\quad u_\mu \coloneqq \prod_i m_i(\mu)!,\] then \(u_\mu^{-1} v_{\lambda\mu}(a) \in \setN[a],\) by the Knop–Sahi formula.

It is convenient to introduce the following notation: Let \(\lambda\) be a diagram, and \(\square\) a box in \(\lambda,\) and define the upper hook length and lower hook length as \[\begin{aligned} h'_\lambda(\square) &\coloneqq a \cdot \arm_\lambda(\square) + \leg_\lambda(\square) + a, \\ h_\lambda(\square) &\coloneqq a \cdot \arm_\lambda(\square) + \leg_\lambda(\square) + 1. \end{aligned}\] These are denoted \(h^*_\lambda(s)\) and \(h_*^\lambda(s),\) respectively, in [Sta89].

Define the two \(a\)-deformations of the product of hook values in the diagram \(\lambda:\) \[H_\lambda = \prod_{\square \in \lambda} h_\lambda(\square), \quad H'_\lambda = \prod_{\square \in \lambda} h'_\lambda(\square).\] The relationship between \(\jackJ_\lambda\) and \(\jackP_\lambda\) is then given by \(\jackJ_\lambda(x;a) = H_\lambda \jackP_\lambda(x;a).\) Furthermore, \[\langle \jackJ_\lambda, \jackJ_\lambda \rangle_a = H_\lambda H'_\lambda \quad \text{and} \quad \langle \jackJ_\mu \jackJ_\nu, \jackJ_\lambda \rangle_a = H_\lambda H_\mu H_\nu \langle \jackP_\mu \jackP_\nu, \jackP_\lambda \rangle_a .\]

#Specializations

The specializations are \[\jackP_\lambda(x;1) = \schurS_\lambda(x),\quad \jackP_{\lambda'}(x;0) = \elementaryE_{\lambda}(x) \text{ and } \jackP_{\lambda}(x;\infty) = \monomial_{\lambda}(x).\]

Similarly, \[\jackJ_\lambda(x;1) = H_\lambda \schurS_\lambda(x), \quad \jackJ_{\lambda'}(x;0) = \lambda! \elementaryE_{\lambda}(x) \text{ and } \jackJ_{\lambda}(x;\infty) = n! \monomial_{\lambda}(x).\]

We also have that \(\jackJ_\lambda(\xvec;2) = \zonal_\lambda(\xvec),\) the zonal symmetric functions.

#Cauchy identity

Recall that \(H_\lambda H'_\lambda = \langle \jackJ_\lambda, \jackJ_\lambda \rangle_a.\) The Cauchy identity for Jack polynomials states that \[\sum_{\lambda} \frac{\jackJ_\lambda(x;a) \jackJ_\lambda(y;a)}{H_\lambda H'_\lambda} = \prod_{i,j} (1-x_i y_j)^{-1/a}.\]

#Calogero–Sutherland and Laplace–Beltrami operators

The Jack polynomials \(\jackJ_\lambda\) are eigenpolynomials for the Calogero–Sutherland operator \[\mathcal{H} \coloneqq \frac{\alpha}{2} \sum_{i=1}^n \left(x_i\frac{\partial}{\partial x_i}\right)^2 + \frac{1}{2} \sum_{i\lt j} \left( \frac{x_i+x_j}{ x_i - x_j } \right) \left( x_i \frac{\partial}{\partial x_i} - x_j \frac{\partial}{\partial x_j} \right),\] so that \[\mathcal{H} \jackJ_\lambda = \sum_{i=1}^n \left( \frac{\alpha}{2} \lambda_i^2 + \frac{n+1-2i}{2} \right) \jackJ_\lambda.\] See [Sut71] for the physics background of \(\mathcal{H}.\) M. Hallnäs constructs a basis for the polynomial eigenfunctions of deformed Calogero–Moser–Sutherland operators [Hal07]. The eigenfunctions are linear combinations of polynomials generalizing super Schur polynomials, and this gives explicit series representations for super Jack polynomials.

They are also eigenpolynomials for the Laplace–Beltrami operator, \[\frac{\alpha}{2} \sum_{i=1}^n \left(x_i\frac{\partial}{\partial x_i}\right)^2 + \frac{1}{2} \sum_{i\neq j} \left( \frac{x_i^2}{ x_i - x_j } \right) \frac{\partial}{\partial x_i}.\] See [LLM99, Rob00] for background.

#Knop–Sahi combinatorial formula

F. Knop and S. Sahi give the following formula for the monomial expansion of the integral form Jack polynomials [KS97]: \[\jackJ_\lambda (x;a) = \sum_{T \in \mathrm{NAF}(\lambda)} d_T(a) x^T\] where \(\mathrm{NAF}(\lambda)\) is the set of non-attacking fillings of the diagram \(\lambda.\) These are non-attacking fillings of \(\lambda\) with natural numbers such that for all boxes \((i,j),\) we have

• \(T(i,j) \neq T(i',j)\) whenever \(i \neq i',\)

• \(T(i,j) \neq T(i',j+1)\) whenever \(i \gt i'.\)

The quantity \(d_T(a)\) is defined as \[d_T(a) = \prod_{s \in crit(T)} [a( \arm_\lambda(s) +1 ) + ( \leg_\lambda(s) +1 )]\] and \(crit(T)\) is the set of boxes \((i,j)\) with \(j\gt{}1\) such that \(T(i,j) = T(i,j-1).\) The Knop–Sahi formula follows from the more general combinatorial formula for Macdonald polynomials in [HHL05].

Y. Naqvi, S. Sahi, and E. Sergel generalize the Knop–Sahi formula to the nonsymmetric interpolation setting [NSS21].

H. Chen and S. Sahi prove monotonicity for the generalized binomial coefficients attached to Schur and Jack binomial formulas [CS25]. As consequences, they obtain Schur-positivity and Jack-positivity inequalities for the corresponding symmetric functions.

#Pieri rule

R. Stanley provides a Pieri rule for Jack polynomials.

The middle product is simply \(r!a^r.\)

G. Shibukawa gives raising-type Pieri formulas for Jack polynomials and applies them to interpolation Jack polynomials [Shi20].

A dual Pieri rule, for computing \(\langle \jackJ_{(1^r)}\jackJ_{(\mu)}, \jackJ_\lambda \rangle,\) is given in [Thm. 6.1, KS96]. It is given for the Jack \(P\) functions and is stated as follows. Let \(X(\lambda/\mu)\) be the set of boxes \((i,j)\) in \(\mu\) such that \(\mu_i=\lambda_i\) and \(\mu'_j \lt \lambda'_j.\) Then \[\langle \jackP_{(1^r)}\jackP_{(\mu)}, \jackP_\lambda \rangle = \prod_{\square \in X(\lambda/\mu)} \frac{ h'_\lambda(\square) h_\mu(\square)}{h_\lambda(\square) h'_\mu(\square)}.\] There is a nice symmetry for the Littlewood–Richardson coefficients, where conjugation of all three shapes corresponds to swapping upper and lower hooks, see [Sec. 2.3, Naq16].

#Jack in power-sum basis

J. Haglund and A. T. Wilson give a formula for \(\jackJ_\lambda (x;a)\) in terms of power-sum symmetric functions [HW17]. It is not cancellation-free in general.

The same paper gives a Schur expansion formula, also not cancellation-free in general.

#Jack Littlewood–Richardson coefficients

Polynomiality of \(g^\lambda_{\mu\nu}(a)\) has been proved [KS97], so only the nonnegativity result remains open. The case when the indexing partitions have at most three parts is proved in [Naq16], and another case involving rectangular shapes is considered in [CJ13]. T. W. Cai and N. Jing also give an iterative realization of general Jack functions from rectangular Jack functions, using Jack vertex operators, and show that products of these vertex operators form a basis of the symmetric functions [CJ13].

This conjecture has a generalization for shifted Jack polynomials.

For some recent progress on the Jack Littlewood–Richardson coefficients, see [Mic23].

#Skew Jack polynomials

No Knop–Sahi analogue is known for skew Jack polynomials, and no combinatorial formula is known for their monomial expansion, even though the coefficients are conjectured to be in \(\setN[a].\) Some observations are proved in [BG22], and this conjecture is closely related to the positivity of \(g^\lambda_{\mu\nu}(a).\)

#Hanlon’s conjecture

#Schur expansion conjecture

Expanding Jack polynomials in terms of Schur functions seems to have a strong connection with rook polynomials. This is explored in [AHW18]. We pose the following conjecture in that paper.

This conjecture has a rich interplay with rook polynomials, and the relationship between the \(c_k(\mu,\lambda)\) and \(b_{n-k}(\mu,\lambda)\) is similar to that of rook hit polynomials and rook polynomials.

#Gessel expansion conjecture

#Double Jack polynomials

L. Lapointe and P. Mathieu define double Jack polynomials through the stable sector of Jack superpolynomials [LM15]. After the fermionic variables are stripped off, the stabilized superpolynomials factor into a product of two ordinary Jack polynomials, with plethystic transformations and different effective parameters. They also derive a Hamiltonian for the double Jack family, involving Calogero–Sutherland terms and generators of an affine \(\widehat{\mathfrak{sl}}_2\) algebra.

#Affine analogues

P. Etingof and A. K. Jr. define affine analogues of Jack polynomials as eigenfunctions of an affine Sutherland operator for the affine root system \(\hat{A}_{n-1}\) [EJ94]. They construct these polynomials explicitly as traces of intertwiners for an affine Lie algebra. Their \(q\)-analogue gives affine analogues of Macdonald polynomials.

#Jack characters

The Jack characters are certain normalizations of the coefficients when Jack polynomials are expanded in the power-sum basis.

Let \(\lambda \vdash n\) and \(\mu \vdash m.\) Then the Jack character (introduced in [Las08, Las09]) is defined as \[\theta_{\mu}^{(\alpha)}(\lambda) \coloneqq \begin{cases} 0 & \text{if } n\lt m, \\ \binom{n- m + m_1(\mu)}{m_1(\mu)} [\powerSum_{\mu,1^{n-m}}]\jackJ_\lambda^{(\alpha)} & \text{if } n\geq m, \end{cases}\] where \(m_1(\mu)\) denotes the number of parts equal to \(1\) in \(\mu.\)

N. Lindzey studies Jack derangements by summing Jack characters over partitions with no singleton parts [Lin23]. At the specializations \(\alpha=1\) and \(\alpha=2,\) these sums recover eigenvalues for the permutation derangement graph and the perfect matching derangement graph, respectively. The same work gives a Jack analogue of a hook-product formula for these derangement sums.

M. Lassalle conjectured that Jack characters satisfy a certain positivity property [Las08]. This was proved by H.B. Dali and M. Dołęga [DD23]. The result is stated in Stanley’s multirectangular coordinates.

Let \(\lambda = [s_1^{r_1}, s_2^{r_2}, s_k^{r_k}]\) be the multi-rectangular coordinates for \(\lambda,\) where \(s_1 \geq s_2 \geq \dotsb \geq s_k.\) That is, \(\lambda\) consists of \(k\) rectangles of size \(s_i \times r_i\) stacked on top of each other. In these coordinates, \(\theta_{\mu}^{(\alpha)}(\mathbf{r},\mathbf{s})\) is a polynomial in \(\setQ[\alpha,s_1,\dotsc,s_k,r_1,\dotsc,r_k].\) R. Stanley conjectured a combinatorial formula in the case \(\alpha=1\) [Sta03], and this was later proved by V. Féray [Fe10].

H.B. Dali and M. Dołęga also give a combinatorial formula for \((-1)^{|\mu|}\theta_{\mu}^{(\alpha)}(\lambda)\) in terms of layered, non-oriented maps.

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