Geometric Realization of Jack Symmetric Functions

• Jack symmetric functions are homogeneous symmetric functions parameterized by α, where their coefficients (Jack characters) encode rich algebraic and geometric information.
• The approach leverages weighted bicolored maps on orientable and non-orientable surfaces to interpret and validate positivity conjectures via explicit combinatorial-topological models.
• Diagrammatic calculi and edge weight assignments provide a concrete framework linking symmetric function theory with random matrix theory, zonal polynomials, and deformation theory.

Jack symmetric functions are a family of homogeneous symmetric functions depending on a parameter $\alpha > 0$, central to algebraic combinatorics and mathematical physics. Their coefficients in the power-sum expansion, known as Jack characters, encode substantial representation-theoretic and geometric information. The geometric realization of Jack symmetric functions refers to expressing these coefficients as weighted counts of combinatorial-topological objects—specifically, bicolored maps drawn on possibly non-orientable surfaces—with the deformation parameter $\alpha$ controlling weights associated to orientable and non-orientable features. This approach yields a topological and combinatorial interpretation of Jack polynomial structure, establishing deep links with random matrix theory, maps enumeration, and free probability.

1. Jack Symmetric Functions and Jack Characters

The Jack symmetric functions $J^{(\alpha)}_\lambda$ are symmetric, homogeneous functions indexed by a partition $\lambda$ and parameterized by $\alpha > 0$. In the standard normalization, their expansion in the power-sum basis is given by: $$J^{(\alpha)}_\lambda = \sum_{\rho \vdash |\lambda|} \theta^{(\alpha)}_\rho(\lambda) \, p_\rho$$ where $p_\rho = \prod_i p_{\rho_i}$ denotes the usual power-sum symmetric functions. The quantity $\theta^{(\alpha)}_\rho(\lambda)$ is the coefficient of $p_\rho$ in $J^{(\alpha)}_\lambda$.

The Jack character $\Ch^{(\alpha)}_\pi(\lambda)$ is a particular renormalization of $$\theta^{(\alpha)}_{\pi \cup 1^{|\lambda| - |\pi|}}(\lambda)$$: $\Ch^{(\alpha)}_{\pi}(\lambda) = \alpha^{-\frac{|\pi| - \ell(\pi)}{2}}\, \binom{|\lambda| - |\pi| + m_1(\pi)}{m_1(\pi)} z_{\pi} \, \theta^{(\alpha)}_{\pi \cup 1^{|\lambda| - |\pi|}}(\lambda)$ where $z_\pi = \prod_i i^{m_i(\pi)} m_i(\pi)!$, $\ell(\pi)$ is the number of parts of $\pi$, and $m_1(\pi)$ is the number of parts of $\pi$ equal to $1$. For $\alpha=1$, one recovers the normalized irreducible characters of the symmetric group (Dołęga et al., 2013).

2. Lassalle’s Positivity Conjectures and the Map Ansatz

Lassalle formulated influential conjectures on Jack characters, notably:

• Stanley–multirectangular positivity: Expressing $\Ch^{(\alpha)}_{\pi}(P \times Q)$ as a polynomial in multirectangular coordinates of the Young diagram and in $\beta = \alpha - 1$ yields nonnegative coefficients.
• Free-cumulant positivity: Expanding $\Ch^{(\alpha)}_\pi$ in anisotropic free cumulant basis produces nonnegative integer coefficients upon appropriate normalization.

These phenomena suggest a combinatorial-topological model—maps on surfaces—such that the enumeration of these maps, possibly weighted, reconstructs Jack characters. A plausible implication is that the algebraic positivity of Jack characters admits a geometric origin in map counting with suitably defined weights that incorporate orientability (Dołęga et al., 2013).

3. Maps on Surfaces and the Definition of Weights

A (possibly disconnected) non-oriented map is a cellular embedding of a bicolored graph onto a (possibly non-orientable) compact surface, such that each complementary region is a topological disk. Each map $M$ can be encoded combinatorially by three pairings $(B, W, E)$ on $2n$ labels (edge-sides).

Associate a deformation parameter: $\gamma(\alpha) = \frac{1 - \alpha}{\sqrt{\alpha}}$ Each edge $e$ of $M$ receives a local weight according to its topological type:

Edge Type	Local Weight $\weight_{M,e}$
Straight edge	$1$
Twisted edge	$\gamma$
Interface edge	$\frac{1}{2}$

Given a linear ordering (history) $\prec$ on edges, edges are removed sequentially, and at each step, the current type of the edge in the evolving map determines the weight. The combined history weight is: $\weight_{M,\prec} = \prod_{i=1}^{|E(M)|} \weight_{M_{i-1}, e_i}$ Averaging over all $n!$ orderings gives the map weight: $w(M; \alpha) = \frac{1}{n!} \sum_{\prec} \weight_{M,\prec}$ The degree in $\gamma$ of $w(M; \alpha)$ is bounded by $2\,\#\text{components of } M - \chi(M)$, where $\chi(M)$ is the Euler characteristic. Twisted removals correspond topologically to cutting along Möbius bands (increasing the non-orientable genus), while straight removals correspond to handle cuts (orientable genus) and interface removals preserve Euler characteristic (Dołęga et al., 2013).

4. The Stanley–Map Formula for Jack Characters

Let $\mathcal{M}(\pi)$ denote all non-oriented maps with face-type $\pi$. The conjectural expansion (proven in numerous settings, including rectangular Young diagrams) expresses the Jack character as a sum over maps: $\Ch^{(\alpha)}_\pi(\lambda) = (-1)^{\ell(\pi)} \sum_{M \in \mathcal{M}(\pi)} (-1)^{|V_\bullet(M)|}\; w(M;\alpha)\; N^{(\alpha)}_M(\lambda)$ where $N^{(\alpha)}_M(\lambda)$ is the “$\alpha$-deformed” number of embeddings of the underlying bicolored graph into the Young diagram $\lambda$,

$$N^{(\alpha)}_M(\lambda) = \alpha^{-\frac{|V_\bullet(M)|}{2}}\, \alpha^{+\frac{|V_\circ(M)|}{2}}\, N_M(\lambda)$$

and $N_M(\lambda)$ is the classical embedding count. For rectangular shapes $\lambda = p \times q$, the embedding number simplifies to $p^{|V_\bullet(M)|} q^{|V_\circ(M)|}$, and map enumeration establishes the positivity of all coefficients in the multirectangular expansion, matching Lassalle’s conjecture in this regime (Dołęga et al., 2013).

5. Geometric Interpretation of Orientability

The weight $w(M; \alpha)$ encodes orientable and non-orientable features:

• Twisted edge removals (“$\gamma$”): Each corresponds to a Möbius band cut, incrementing non-orientable genus.
• Straight edge removals: Correspond to handle addition (genus increment on an orientable surface).
• Interface edges: Merge faces without affecting Euler characteristic.

For $\alpha = 1$ ($\gamma = 0$), only orientable maps contribute, recovering the Schur character expansion. For $\alpha = 2$ or $1/2$, $\gamma = \pm 1/\sqrt{2}$ and the model specializes to known zonal polynomial expansions for all non-oriented maps, weighted by powers of $\alpha$ corresponding to vertex types. The polynomial structure of $w(M;\alpha)$ as a function of $\gamma$ links directly to the non-orientable genus content of a map (Dołęga et al., 2013).

6. Diagrammatic and Graphical Calculi for Jack Structures

The Jack inner product

$$\langle p_\lambda, p_\mu \rangle_\alpha = \delta_{\lambda, \mu} z_\lambda \alpha^{\ell(\lambda)}$$

admits a graphical realization via the annular action of $B$-decorated string diagrams, where $B$ is a finite-dimensional $\mathbb{Z}$-graded Frobenius superalgebra with $\alpha = \dim B_\text{even} - \dim B_\text{odd}$ parameterizing the Jack deformation. The graphical calculus establishes an explicit isomorphism: $$\Phi: \mathrm{Tr}(H_B^*)^+ \to \mathrm{Sym} \quad (P_n^+ \mapsto p_n)$$ such that diagrammatic pairings reproduce the Jack inner product. This graphical identification clarifies the algebraic origins of Jack structural constants and demonstrates their geometric nature inside the framework of Heisenberg algebra and category theory (Licata et al., 2016).

7. Special Cases, Positivity, and Example Calculations

For rectangular Young diagrams, the combinatorics of embedding bicolored maps simplifies, and direct combinatorial bijections demonstrate that Jack characters are given by positive sums over weighted maps, validating multirectangular positivity for these diagrams. Diagrammatic calculations (e.g., pairing annular diagrams corresponding to partitions) quantitatively recover the structure constants of the Jack inner product. For instance, the pairing for partition $(2,1)$ yields $2\alpha^2$, matching the combinatorics of the associated diagrams and their algebraic images in the symmetric function ring (Dołęga et al., 2013, Licata et al., 2016).

The geometric realization of Jack symmetric functions thus links algebraic, combinatorial, and topological structure, manifesting connections among symmetric group representations, map enumeration, and deformation theory via parameter $\alpha$. The approach not only leads to deep conjectures (like Lassalle's) regarding positivity and combinatorial meaning but provides explicit machinery for calculating symmetric function coefficients in geometric and diagrammatic terms.