Differential equations for the series of hypermaps with control on their full degree profile

TL;DR

This work introduces and proves a differential-equation framework for the full generating series of hypermaps with controlled vertex and face degrees, including α-deformations tied to Jack polynomials. By leveraging Jack characters and a family of differential operators, the authors derive a central equation for the three-alphabet generating series $G^{(α)}$ and extend it to constellations and Hurwitz-type problems with full ramification profiles. They provide a detailed combinatorial interpretation in the α=1 case, establish connections to the Matching–Jack conjecture, and supply explicit formulas for structure coefficients $g^oldsymbol{ ho}_{oldsymbol{ u},oldsymbol{ au}}(α)$ in terms of operator-derived quantities $a^oldsymbol{ au}_oldsymbol{ u}$ and $d^oldsymbol{ au}_{oldsymbol{ u},oldsymbol{ au}}$, along with a polynomiality/positivity discussion. The results yield differential equations that characterize the generating series and bridge algebraic combinatorics, symmetric-function theory, and enumerative geometry of maps and Hurwitz numbers, with a clean path to connected-series formulations. Overall, the paper provides a robust, general mechanism to count hypermaps with detailed ramification data and to study their α-deformations within a unified operator-theoretic and combinatorial framework.

Abstract

We consider the generating series of oriented and non-oriented hypermaps with controlled degrees of vertices, hyperedges and faces. It is well known that these series have natural expansions in terms of Schur and Zonal symmetric functions, and with some particular specializations, they satisfy the celebrated KP and BKP equations. We prove that the full generating series of hypermaps satisfy a family of differential equations. We give a first proof which works for an $α$ deformation of these series related to Jack polynomials. This proof is based on a recent construction formula for Jack characters using differential operators. We also provide a combinatorial proof for the orientable case. Our approach also applies to the series of $k$-constellations with control of the degrees of vertices of all colors. In other words, we obtain an equation for the generating function of Hurwitz numbers (and their $α$-deformations) with control of full ramification profiles above an arbitrary number of points. Such equations are new even in the orientable case.

Figures (7)

• Figure 1: An example of a vertex labelled hypermap of profile $([3,2,2],[5,2],[6,1])$. Faces of color $(-)$ are represented in blue, the root of the vertex $v_{d,i}$ is denoted by $\color{blue}c_{d,i}$.
• Figure 2: Example of the action of $\mathcal{C}^{(1)}_2$ on a map $N$. On the left the map $N$, and on the right a map $M$ obtained by adding one black vertex $v$, two white vertices $w_1$ and $w_2$ and 6 edges (represented in blue).
• Figure 3: Types of bicolor edges in a BFC map.
• Figure 4: Deleting edges of a hypermap with marked faces to obtain a pre-hypermap.
• Figure 5: On the left a labelled hypermap with one marked face; $(-)$ faces are represented in blue and the marked face is crossed. On the right the associated pre-hypermap.

Theorems & Definitions (73)

• Definition 1.1
• Definition 1.2
• Theorem 1.3: BenDaliDolega2023
• Proposition 1.4
• Theorem 1.5
• Theorem 1.6
• Conjecture 1
• Theorem 1.7
• Corollary 1.8
• Theorem 2.1: Macdonald1995