Jack Polynomials in Algebraic Combinatorics

• Jack polynomials are a one-parameter family of symmetric functions defined by triangularity and orthogonality, serving as a cornerstone in algebraic combinatorics and representation theory.
• Their eigenfunction property for the Calogero–Sutherland operator enables precise analysis of quantum integrable systems and clustering phenomena.
• Generalizations including vector-valued, prescribed symmetry, and supersymmetric Jack polynomials expand their utility in random matrix theory and conformal field theory.

Jack polynomials are a one-parameter family of symmetric functions in several variables, uniquely determined by orthogonality and triangularity properties with respect to dominance order. They are central objects in algebraic combinatorics, representation theory of symmetric groups and related algebras, integrable systems (especially the quantum Calogero–Sutherland model), random matrix theory, and the theory of symmetric functions. Their rich structure is reflected in their connections to special cases (Schur polynomials, zonal polynomials), their encompassing of connection coefficients, and their generalizations to vector-valued settings, prescribed symmetries, and supersymmetric extensions.

1. Definition, Characterization, and Normalization

Let $N$ be a positive integer, and $\mathcal{P}_N$ the set of partitions $$\lambda = (\lambda_1 \geq \lambda_2 \geq \ldots \geq \lambda_N \geq 0)$$. For a parameter $\alpha > 0$, the (scalar) Jack polynomial $P_\lambda^{(\alpha)}(x_1,\ldots,x_N)$ is the unique homogeneous symmetric polynomial of degree $|\lambda| = \sum_i \lambda_i$ in $N$ variables, satisfying:

• Triangularity: $$P_\lambda^{(\alpha)} = m_\lambda + \sum_{\mu < \lambda} c_{\lambda \mu}(\alpha) m_\mu$$, where $m_\mu$ is the monomial symmetric function and $<$ denotes dominance order.
• Orthogonality: Jack polynomials are orthogonal with respect to the inner product

$\mathcal{P}_N$0

where $\mathcal{P}_N$1 are the power-sum symmetric functions, $\mathcal{P}_N$2, and $\mathcal{P}_N$3 is the length of $\mathcal{P}_N$4.

• Eigenfunction Property: Jack polynomials diagonalize a quantum Calogero–Sutherland (or Laplace–Beltrami or Sutherland) operator:

$\mathcal{P}_N$5

with

$\mathcal{P}_N$6

The eigenvalue $\mathcal{P}_N$7 is an explicit function of $\mathcal{P}_N$8 and $\mathcal{P}_N$9.

Normalization conventions vary. The “monic” convention is most standard: the coefficient of $$\lambda = (\lambda_1 \geq \lambda_2 \geq \ldots \geq \lambda_N \geq 0)$$0 in $$\lambda = (\lambda_1 \geq \lambda_2 \geq \ldots \geq \lambda_N \geq 0)$$1 is $$\lambda = (\lambda_1 \geq \lambda_2 \geq \ldots \geq \lambda_N \geq 0)$$2 (Lapointe et al., 2015, Desrosiers et al., 2013). Sometimes “normed” versions or versions normalized at $$\lambda = (\lambda_1 \geq \lambda_2 \geq \ldots \geq \lambda_N \geq 0)$$3 are used (Shibukawa, 2020).

In limiting cases, Jack polynomials specialize:

• $$\lambda = (\lambda_1 \geq \lambda_2 \geq \ldots \geq \lambda_N \geq 0)$$4 to Schur polynomials,
• $$\lambda = (\lambda_1 \geq \lambda_2 \geq \ldots \geq \lambda_N \geq 0)$$5 to zonal polynomials,
• $$\lambda = (\lambda_1 \geq \lambda_2 \geq \ldots \geq \lambda_N \geq 0)$$6 to the monomial symmetric functions (Desrosiers et al., 2013, Shibukawa, 2020).

2. Combinatorial and Structural Properties

Jack polynomials admit several explicit bases expansions:

• Monomial Expansion: $$\lambda = (\lambda_1 \geq \lambda_2 \geq \ldots \geq \lambda_N \geq 0)$$7 is monic triangular in this basis.
• Power-sum Expansion:

$$\lambda = (\lambda_1 \geq \lambda_2 \geq \ldots \geq \lambda_N \geq 0)$$8

with Jack characters $$\lambda = (\lambda_1 \geq \lambda_2 \geq \ldots \geq \lambda_N \geq 0)$$9, providing a link to the representation theory of the symmetric group (Vassilieva, 2013, Dołęga et al., 2012).

Orthogonality: The normalization factor for $\alpha > 0$0 is $\alpha > 0$1, where the product is over boxes $\alpha > 0$2 in the diagram of $\alpha > 0$3, and $\alpha > 0$4 denote the arm and leg lengths, respectively:

$\alpha > 0$5

These yield the orthogonality normalization (Vassilieva, 2013).

Product and Structure Constants: The product of two Jack polynomials expands linearly in terms of Jacks:

$\alpha > 0$6

where the structure constants $\alpha > 0$7 play the role of (Jack) connection coefficients, interpolating between classical class algebra and zonal algebra structure constants as $\alpha > 0$8 (Vassilieva, 2013). These coefficients admit both direct combinatorial and representation-theoretic interpretations.

Formulas for Binomial Coefficients: The "binomial coefficients" for Jack polynomials, i.e., the expansion coefficients of $\alpha > 0$9 in Jack polynomial basis, decompose as a product of "stem" and "leaf" factors, with explicit combinatorial formulas involving hook lengths and the detailed structure of the skew shape $P_\lambda^{(\alpha)}(x_1,\ldots,x_N)$0 (Naqvi et al., 2018). In the two-row case, a symmetry $P_\lambda^{(\alpha)}(x_1,\ldots,x_N)$1 (row exchange) is revealed in the explicit formula for the "leaf" factor.

3. Generalizations: Prescribed Symmetry, Vector-Valued, and Supersymmetric Jacks

Jack Polynomials with Prescribed Symmetry: For $P_\lambda^{(\alpha)}(x_1,\ldots,x_N)$2 variables partitioned into two blocks, one can define Jack polynomials that are anti-symmetric in the first $P_\lambda^{(\alpha)}(x_1,\ldots,x_N)$3 variables and symmetric in the remaining $P_\lambda^{(\alpha)}(x_1,\ldots,x_N)$4—the AS-type (Desrosiers et al., 2013). These arise naturally via symmetrization and antisymmetrization of nonsymmetric Jacks, using superpartitions as indices, and have applications in both algebraic combinatorics and the fractional quantum Hall effect.

Vector-Valued Jack Polynomials: For any irreducible $P_\lambda^{(\alpha)}(x_1,\ldots,x_N)$5-module $P_\lambda^{(\alpha)}(x_1,\ldots,x_N)$6, one can construct polynomials

$P_\lambda^{(\alpha)}(x_1,\ldots,x_N)$7

as simultaneous eigenfunctions of Dunkl or Cherednik–Dunkl operators, forming a basis of so-called standard modules for Cherednik algebras (Dunkl et al., 2010, Dunkl, 2018, Dunkl, 2020). These vector-valued Jacks generalize symmetric Jack polynomials: when $P_\lambda^{(\alpha)}(x_1,\ldots,x_N)$8 is trivial (resp. sign), one recovers the usual (alternating) scalar Jacks (Dunkl, 2010). Explicit norm formulas generalizing the classical hook-length formula exist for these vector-valued cases.

Supersymmetric and Double Jack Polynomials: In the supersymmetric case, the theory encompasses polynomial rings with both commuting ($P_\lambda^{(\alpha)}(x_1,\ldots,x_N)$9) and anticommuting ($|\lambda| = \sum_i \lambda_i$0) variables (Lapointe et al., 2015). For sufficiently large fermionic degree, Jack superpolynomials stabilize—resulting in "double Jack" polynomials, factorized as products of ordinary Jacks with shifted parameters in different (bosonic) variables. These double Jacks are eigenfunctions of an explicit integrable Hamiltonian involving an $|\lambda| = \sum_i \lambda_i$1 structure.

4. Pieri-Type and Binomial Formulas, Clustering, and Special Parameter Values

Pieri Formulas: Classical Pieri rules express the product of a Jack polynomial by an elementary symmetric function or complete symmetric function as a sum over Jack polynomials with easily described index sets (Shibukawa, 2020, Shibukawa, 2020). Twisted ("raising-type" or "falling-type") Pieri formulas extend this to more general differential or difference-operator situations, underpinning interpolation and shifted Jack polynomials and their applications to difference equations.

Clustering and Admissible Partitions: For special negative values of $|\lambda| = \sum_i \lambda_i$2 (nontrivial in algebraic applications and quantum Hall physics), Jack polynomials indexed by so-called admissible partitions (e.g., those satisfying $|\lambda| = \sum_i \lambda_i$3 for all relevant $|\lambda| = \sum_i \lambda_i$4) have vanishing properties—specifically, they vanish to order $|\lambda| = \sum_i \lambda_i$5 when $|\lambda| = \sum_i \lambda_i$6 variables coincide ("clustering") (Blondeau-Fournier et al., 2016, Desrosiers et al., 2013). Feigin–Jimbo–Miwa–Mukhin and others have shown that clustering is tightly governed by these admissibility conditions. This mechanism is essential in the vertex operator algebra approach to conformal field theory and the classification of modules for superconformal algebras.

5. Positivity, Integrality, and Map Enumeration: Lassalle's Conjecture and Jack Characters

Jack Characters and Combinatorics: The coefficients $|\lambda| = \sum_i \lambda_i$7 in the power-sum expansion are called Jack characters. They generalize irreducible characters of symmetric groups and zonal spherical functions (Vassilieva, 2013, Dołęga et al., 2012). Recent results provide explicit positive combinatorial formulas for these characters in terms of statistics of bipartite maps, ribbon paths, and other structures (Dali et al., 2023).

Lassalle’s Positivity–Integrality Conjecture: Lassalle conjectured that the coefficients of Jack characters, when expressed in Stanley's multirectangular coordinates, are polynomials in $|\lambda| = \sum_i \lambda_i$8 with nonnegative integer coefficients. This has now been proven: the Jack character admits a manifestly positive map-sum expansion, confirming the integrality and nonnegativity (Dali et al., 2023).

Kerov Polynomials: Jack characters can be expressed as polynomials in certain free-cumulant–like functionals (Kerov polynomials), with coefficients polynomial in $|\lambda| = \sum_i \lambda_i$9, satisfying degree constraints and strong positivity properties (Dołęga et al., 2012).

Connection Coefficients and the Matchings–Jack Conjecture: The connection coefficients $N$0 interpolate, as functions of $N$1, between central structure constants of the symmetric group ($N$2) and of the double coset algebra ($N$3). The Matchings–Jack conjecture posits that $N$4 are polynomials in $N$5 with nonnegative integer coefficients, realized as weighted sums over perfect matchings, with the weight enumerating "non-orientability" or "non-bipartiteness" (Vassilieva, 2013). This remains open in general but is settled in many infinite families.

6. Representation-Theoretic and Geometric Aspects

Module Theory over Cherednik Algebras: Jack polynomials, especially in the vector-valued or non-symmetric settings, are fundamental in the theory of rational Cherednik algebras. The module structure, contravariant forms, and singular values (determined by hook lengths) provide detailed information on unitarity loci, positivity of inner products, and the support of module constituents (Dunkl, 2018, Dunkl et al., 2010).

Generalized Jack Polynomials: In geometric representation theory, generalized Jacks arise as canonical bases in the equivariant cohomology of instanton moduli spaces, as images of fixed-point classes under stable envelopes (Smirnov, 2014). These admit explicit expansions in terms of multi-Schur functions, with combinatorics controlled by hyperplane arrangements and stable map data.

7. Applications and Further Directions

• Random Matrix Theory: Jack polynomials and related functions provide explicit formulas for the densities and moments of multivariate distributions, including singular beta-Wishart matrices, playing a key role in random matrix theory/statistical hypothesis testing (Shimizu et al., 2021, Diaz-Garcia et al., 2010).
• Quantum Integrable Systems: As eigenfunctions of Calogero–Sutherland and related models, Jacks model the quantum behavior of multi-particle systems with inverse-square interactions (Lapointe et al., 2015, Desrosiers et al., 2013).
• Combinatorial Map Enumeration: Expansions in Jack characters enumerate maps of specified types, linking algebraic properties of Jacks to enumerative geometry and topological graph theory (Dołęga et al., 2012, Vassilieva, 2013, Dali et al., 2023).
• Vertex Operator Algebras and CFT: Admissible Jack polynomials at special negative values of the parameter govern the structure and module classification for rational (super)conformal field theories via Zhu algebras (Blondeau-Fournier et al., 2016).

Emerging directions include the extension of positivity and combinatorial structure to Macdonald and Koornwinder polynomials and multivariate binomial-type families, expansions of Jack connection coefficients, and deeper connections between representation theory, probability, and integrable systems.

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Key References

• Jack polynomials with prescribed symmetry and clustering: (Desrosiers et al., 2013)
• Superconformal models and admissible Jack polynomials: (Blondeau-Fournier et al., 2016)
• Vector-valued Jack polynomials: (Dunkl et al., 2010, Dunkl, 2018, Dunkl, 2010, Dunkl, 2020)
• Positivity, map enumeration, and Lassalle’s conjecture: (Dali et al., 2023)
• Pieri formulas and interpolation: (Shibukawa, 2020, Shibukawa, 2020)
• Binomial formula combinatorics: (Naqvi et al., 2018)
• Polynomiality and Kerov polynomials: (Dołęga et al., 2012)
• Generalized Jack polynomials and equivariant cohomology: (Smirnov, 2014)
• Supersymmetric Jacks and double Jacks: (Lapointe et al., 2015)
• Connection coefficients, combinatorics, and the Matchings–Jack conjecture: (Vassilieva, 2013)