# Jack function

A **Jack function** (more commonly, Jack polynomial) is a symmetric polynomial \( J^{(\alpha)}_{\lambda} \) in a set of variables, indexed by an integer partition \( \lambda \) and a positive deformation parameter \( \alpha \), that is orthogonal with respect to a parameter-dependent inner product on symmetric functions.<sup>[1](https://mathworld.wolfram.com/JackPolynomial.html)</sup><sup> • </sup><sup>[2](https://people.math.harvard.edu/~bendali/Thesis_Dissertation.pdf)</sup> The family interpolates between several classical bases of symmetric function theory: at \( \alpha = 1 \) it gives the [Schur functions](https://www.edgepedia.org/schur-functions), at \( \alpha = 2 \) the zonal polynomials, and at \( \alpha = 1/2 \) the quaternionic zonal polynomials.<sup>[1](https://mathworld.wolfram.com/JackPolynomial.html)</sup>

| Key fact | Detail |
|---|---|
| Definition | Symmetric functions \( J^{(\alpha)}_{\lambda} \) indexed by a partition \( \lambda \) and a parameter \( \alpha \), uniquely characterized by orthogonality, triangularity in the dominance order, and a normalization condition<sup>[2](https://people.math.harvard.edu/~bendali/Thesis_Dissertation.pdf)</sup> |
| Special cases | \( \alpha = 1 \): Schur polynomials; \( \alpha = 2 \): zonal polynomials; \( \alpha = 1/2 \): quaternion zonal polynomials<sup>[1](https://mathworld.wolfram.com/JackPolynomial.html)</sup> |
| Origin | Defined by Henry Jack in 1969–1970 while evaluating an integral connected with the noncentral Wishart distribution<sup>[1](https://mathworld.wolfram.com/JackPolynomial.html)</sup> |
| Defining operator | Eigenfunctions of the Laplace–Beltrami operator \( D^{(\alpha)} \) on symmetric functions<sup>[2](https://people.math.harvard.edu/~bendali/Thesis_Dissertation.pdf)</sup> |
| Combinatorial formula | Monomial expansion coefficients interpreted combinatorially by Knop and Sahi (1997), answering a question of Foulkes (1974)<sup>[1](https://mathworld.wolfram.com/JackPolynomial.html)</sup> |
| Applications | Random matrix theory, multivariate statistics, Selberg integrals, and the Calogero–Sutherland model of quantum mechanics<sup>[1](https://mathworld.wolfram.com/JackPolynomial.html)</sup><sup> • </sup><sup>[2](https://people.math.harvard.edu/~bendali/Thesis_Dissertation.pdf)</sup> |
| Software | Available in Maple (MOPs), SageMath, and the R packages jack and jackpolynomials<sup>[3](https://dspace.mit.edu/entities/publication/049e75ba-82a4-4d81-8e50-6055c3c0d219)</sup><sup> • </sup><sup>[4](https://github.com/stla/jackpolynomials)</sup> |

## Definition and characterization

Macdonald established a characterization theorem that is commonly taken as the definition. The Jack polynomials \( (J^{(\alpha)}_{\lambda})_{\lambda} \) are the unique family of symmetric functions indexed by partitions satisfying three properties: orthogonality, \( \langle J^{(\alpha)}_{\lambda}, J^{(\alpha)}_{\mu} \rangle_{\alpha} = 0 \) for \( \lambda \neq \mu \); triangularity, meaning \( J^{(\alpha)}_{\lambda} \) expands in the monomial basis \( m_{\nu} \) only over partitions \( \nu \) dominated by \( \lambda \) in the dominance order; and the normalization \( [p_1^{\,n}] J^{(\alpha)}_{\lambda} = 1 \), fixing the coefficient of \( p_1^{\,n} \) in the power-sum expansion. The family forms a basis of the space \( S_{\alpha} \) of symmetric functions.<sup>[2](https://people.math.harvard.edu/~bendali/Thesis_Dissertation.pdf)</sup> [Orthogonality](https://www.edgepedia.org/orthogonality) of the Jack polynomials is proved in Macdonald's 1995 book, page 383.<sup>[1](https://mathworld.wolfram.com/JackPolynomial.html)</sup>

An equivalent characterization runs through differential operators. Jack polynomials are the eigenfunctions of the [Laplace–Beltrami operator](https://www.edgepedia.org/laplace-beltrami-operator) \( D^{(\alpha)} \), with eigenvalue the sum of the \( \alpha \)-contents \( c_{\alpha}(\square) \) of the cells of the [Young diagram](https://www.edgepedia.org/young-diagram) of \( \lambda \).<sup>[2](https://people.math.harvard.edu/~bendali/Thesis_Dissertation.pdf)</sup> This operator perspective connects the polynomials to the Calogero–[Sutherland](https://www.edgepedia.org/sutherland) integrable system of quantum mechanics.<sup>[2](https://people.math.harvard.edu/~bendali/Thesis_Dissertation.pdf)</sup>

Several normalizations are in use. The three common ones are called "C," "J," and "P": the "J" normalization makes the coefficient of the lowest-order monomial equal to \( n! \), the "P" normalization is monic, and the "C" normalization is fixed by the condition that summing \( C_{\kappa}^{(\alpha)} \) over partitions \( \kappa \) of \( k \) recovers \( (x_1 + \cdots + x_n)^k \).<sup>[1](https://mathworld.wolfram.com/JackPolynomial.html)</sup> The first few "J" polynomials are \( J^{\alpha}_{[1]} = m_{[1]} \), \( J^{\alpha}_{[2]} = (1+\alpha) m_{[2]} + 2 m_{[1,1]} \), and \( J^{\alpha}_{[1,1]} = 2 m_{[1,1]} \).<sup>[1](https://mathworld.wolfram.com/JackPolynomial.html)</sup>

A tableau formula gives the "P" normalization directly: \( P_{\mu}(x; a) \) is a sum over reverse semistandard tableaux of shape \( \mu \), each weighted by a rational function \( \psi_T(a) \) built from arm and leg lengths of the boxes of the shape.<sup>[5](https://www.symmetricfunctions.com/jack.htm)</sup>

## History

The polynomials arose in statistics. Henry Jack, working in the late 1960s, was attempting to evaluate an integral connected with the noncentral [Wishart distribution](https://www.edgepedia.org/wishart-distribution), and in the course of that work defined a set of polynomials, one for each partition of an integer, that are invariant under the [orthogonal group](https://www.edgepedia.org/orthogonal-group) and depend on a real parameter \( \alpha \).<sup>[1](https://mathworld.wolfram.com/JackPolynomial.html)</sup><sup> • </sup><sup>[6](https://www.cambridge.org/core/journals/proceedings-of-the-royal-society-of-edinburgh-section-a-mathematics/article/abs/ia-class-of-symmetric-polynomials-with-a-parameter/8BA8AF9C71233500191ABBDC576C35D2)</sup> In his 1970 paper in the Proceedings of the Royal Society of Edinburgh he observed that at \( \alpha = -1 \) the polynomials coincide with the augmented monomial symmetric functions, that a systematic way of taking linear combinations at \( \alpha = 1 \) recovers the Schur functions of symmetric-group representation theory, and that a similar procedure at \( \alpha = 2 \) gives the zonal polynomials defined by James in 1964.<sup>[6](https://www.cambridge.org/core/journals/proceedings-of-the-royal-society-of-edinburgh-section-a-mathematics/article/abs/ia-class-of-symmetric-polynomials-with-a-parameter/8BA8AF9C71233500191ABBDC576C35D2)</sup> Jack noted the \( \alpha = 1 \) case himself and conjectured that \( \alpha = 2 \) gives the zonal polynomials.<sup>[1](https://mathworld.wolfram.com/JackPolynomial.html)</sup>

Later authors placed the family in a larger setting. Stanley (1989) and Macdonald (1995) generalized many known properties of the Schur and zonal polynomials to Jack polynomials.<sup>[1](https://mathworld.wolfram.com/JackPolynomial.html)</sup> Around 1960 two different one-parameter deformations of the Schur polynomials had been suggested, and Macdonald united them into a two-parameter deformation now called the [Macdonald polynomials](https://www.edgepedia.org/macdonald-polynomials); the Jack polynomials correspond to the limit regime \( t = q^{\theta} \to 1 \) with \( \theta > 0 \), while the Hall–Littlewood polynomials arise at \( q = 0 \).<sup>[7](https://www.mathunion.org/fileadmin/ICM/Proceedings/ICM2014.1/ICM2014.1.pdf)</sup> In the same direction, the Jack polynomials extrapolate the zonal spherical functions of harmonic analysis on Riemannian [symmetric spaces](https://www.edgepedia.org/symmetric-space) from three distinguished parameter values, corresponding to spaces over the real numbers, the complex numbers, and the quaternions.<sup>[7](https://www.mathunion.org/fileadmin/ICM/Proceedings/ICM2014.1/ICM2014.1.pdf)</sup>

## Specializations and structure constants

The parameter values \( \alpha = 1 \) and \( \alpha = 2 \) recover the two best-studied classical families. At \( \alpha = 1 \) one has \( J^{(1)}_{\lambda} = H_{\lambda} s_{\lambda} \), where \( s_{\lambda} \) is the Schur function and \( H_{\lambda} \) the hook product; at \( \alpha = 2 \) one has \( J^{(2)}_{\lambda} = Z_{\lambda} \), the zonal polynomial. Schur and zonal functions are tied to random matrix theory and to zonal spherical functions in the complex and real cases respectively.<sup>[2](https://people.math.harvard.edu/~bendali/Thesis_Dissertation.pdf)</sup> The squared norm \( j^{(\alpha)}_{\lambda} = \langle J^{(\alpha)}_{\lambda}, J^{(\alpha)}_{\lambda} \rangle_{\alpha} \) has a combinatorial formula as a product of two \( \alpha \)-deformed hook factors, reducing at \( \alpha = 1 \) and \( \alpha = 2 \) to squared hook products.<sup>[2](https://people.math.harvard.edu/~bendali/Thesis_Dissertation.pdf)</sup>

Macdonald also gave a combinatorial formula for the specialization to an alphabet \( u = (u, u, \ldots) \): \( J^{(\alpha)}_{\lambda}(u) \) factors as a product over the cells of \( \lambda \) of terms \( (u + c_{\alpha}(\square)) \).<sup>[2](https://people.math.harvard.edu/~bendali/Thesis_Dissertation.pdf)</sup>

## Combinatorics

The question of a combinatorial interpretation for the polynomials was raised by Foulkes in 1974 and answered by Knop and Sahi in 1997, who interpreted the coefficients of \( J^{(\alpha)}_{\lambda} \) in the monomial basis in terms of tableaux of shape \( \lambda \).<sup>[1](https://mathworld.wolfram.com/JackPolynomial.html)</sup><sup> • </sup><sup>[2](https://people.math.harvard.edu/~bendali/Thesis_Dissertation.pdf)</sup> Haglund and Wilson later obtained formulas for the expansion of Jack polynomials in Schur functions and power-sum functions, given in terms of inversion graphs of tableaux counted with \( \alpha \)-deformed hook weights.<sup>[2](https://people.math.harvard.edu/~bendali/Thesis_Dissertation.pdf)</sup>

The power-sum expansion of Jack polynomials connects to the enumeration of maps. Hanlon conjectured a weight function \( w(\sigma, \tau) \) that would express \( J_{\lambda}(x; a) \) as a signed sum over the row- and column-stabilizers of a standard [Young tableau](https://www.edgepedia.org/young-tableau), and Chapuy and Dołęga showed that, for a related deformation, the resulting coefficients turn out to be polynomial in the deformation parameter \( b \), with integer coefficients that are all nonnegative, and can be understood via generalized branched coverings of both orientable and non-orientable surfaces.<sup>[5](https://www.symmetricfunctions.com/jack.htm)</sup> Skew Jack polynomials remain less complete: no Knop–Sahi analog is known for them, and no combinatorial formula is known for their monomial expansion, although the coefficients are conjectured to lie in \( \mathbb{N}[a] \).<sup>[5](https://www.symmetricfunctions.com/jack.htm)</sup>

## Applications

**Random matrices and statistics.** Jack polynomials are especially useful in the theory of random matrices.<sup>[1](https://mathworld.wolfram.com/JackPolynomial.html)</sup> The \( \beta \)-ensemble distributions, which generalize the classical Gaussian and Wishart ensembles, have been studied for the particular values \( \beta = 1, 2, 4 \) as joint eigenvalue densities for random matrices with real, complex, and quaternion entries; eigenvalue statistics expressible through integrals of symmetric polynomials can be computed in terms of multivariate orthogonal polynomials, including the Jack polynomials.<sup>[3](https://dspace.mit.edu/entities/publication/049e75ba-82a4-4d81-8e50-6055c3c0d219)</sup> In multivariate statistics, the hypergeometric functions of matrix argument introduced by Herz and expanded in zonal polynomials by [Constantine](https://www.edgepedia.org/constantine) underlie the theory of non-central distributions, and Macdonald proposed a one-parameter generalization of this theory replacing the zonal polynomials by Jack polynomials; many results in that general theory remain conjectural, and Macdonald's widely circulated manuscript remains unpublished.<sup>[8](https://www.mathematics.pitt.edu/node/1830)</sup>

**Hypergeometric functions and Selberg integrals.** Kaneko introduced a class of hypergeometric functions of several variables using Jack polynomials and gave a multivariate generalization of Aomoto's result, in work published in the SIAM Journal on Mathematical Analysis in 1993 (volume 24, pages 1086–1110).<sup>[9](https://epubs.siam.org/doi/abs/10.1137/0524064)</sup> The associated hypergeometric series \( {}_p\mathcal{F}^{(\alpha)}_q \) are built from the Jack polynomials together with \( \alpha \)-deformed hook lengths and Pochhammer symbols.<sup>[10](https://doi.org/10.1016/j.jat.2016.11.005)</sup>

**Physics.** Beyond the Calogero–Sutherland model, Jack polynomials appear in statistical mechanics and probability through \( \beta \)-ensembles, generalizations of Selberg integrals, and random partition models.<sup>[2](https://people.math.harvard.edu/~bendali/Thesis_Dissertation.pdf)</sup>

## Computation

The polynomials can be generated in practice from the Macdonald characterization by a Gram–Schmidt orthogonalization process.<sup>[2](https://people.math.harvard.edu/~bendali/Thesis_Dissertation.pdf)</sup> The MOPs Maple library implements algorithms for computing the Jack, Hermite, Laguerre, and Jacobi multivariate polynomials for arbitrary parameters.<sup>[3](https://dspace.mit.edu/entities/publication/049e75ba-82a4-4d81-8e50-6055c3c0d219)</sup> In R, the packages jack and jackpolynomials compute Jack, zonal, Schur, Hall–Littlewood, and Macdonald polynomials, including symbolic Jack polynomials with the parameter left indeterminate.<sup>[4](https://github.com/stla/jackpolynomials)</sup><sup> • </sup><sup>[11](https://search.r-project.org/CRAN/refmans/jack/html/00Index.html)</sup>

## References

1. [Jack Polynomial, Wolfram MathWorld](https://mathworld.wolfram.com/JackPolynomial.html)
2. [H. B. Dali, thesis, Section 1.2: Symmetric functions and Jack polynomials, Harvard University](https://people.math.harvard.edu/~bendali/Thesis_Dissertation.pdf)
3. [Eigenvalue statistics for beta-ensembles, MIT DSpace](https://dspace.mit.edu/entities/publication/049e75ba-82a4-4d81-8e50-6055c3c0d219)
4. [stla/jackpolynomials: Jack, zonal, Schur, and other symmetric polynomials, GitHub](https://github.com/stla/jackpolynomials)
5. [Jack polynomials, SymCat](https://www.symmetricfunctions.com/jack.htm)
6. [H. Jack, A class of symmetric polynomials with a parameter, Proceedings of the Royal Society of Edinburgh Section A: Mathematics](https://www.cambridge.org/core/journals/proceedings-of-the-royal-society-of-edinburgh-section-a-mathematics/article/abs/ia-class-of-symmetric-polynomials-with-a-parameter/8BA8AF9C71233500191ABBDC576C35D2)
7. [ICM 2014 Proceedings, Volume 1](https://www.mathunion.org/fileadmin/ICM/Proceedings/ICM2014.1/ICM2014.1.pdf)
8. [Macdonald conjectures for multivariate hypergeometric functions, University of Pittsburgh Department of Mathematics](https://www.mathematics.pitt.edu/node/1830)
9. [J. Kaneko, Selberg Integrals and Hypergeometric Functions Associated with Jack Polynomials, SIAM Journal on Mathematical Analysis 24 (1993), 1086–1110](https://epubs.siam.org/doi/abs/10.1137/0524064)
10. [Limits for circular Jacobi beta-ensembles, Journal of Approximation Theory](https://doi.org/10.1016/j.jat.2016.11.005)
11. [R: Jack, Zonal, Schur, and Other Symmetric Polynomials, CRAN](https://search.r-project.org/CRAN/refmans/jack/html/00Index.html)

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*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Numbers and algebra › Arithmetic and number systems › Integer sequences and partitions › Partitions › Partitions and symmetric function theory*

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