Primitive decompositions of Johnson graphs

A transitive decomposition of a graph is a partition of the edge set together with a group of automorphisms which transitively permutes the parts. In this paper we determine all transitive decompositions of the Johnson graphs such that the group preserving the partition is arc-transitive and acts primitively on the parts.Comment: 35 page

Triply mixed coverings of arbitrary base curves: Quasimodularity, quantum curves and a mysterious topological recursions

Simple Hurwitz numbers enumerate branched morphisms between Riemann surfaces with fixed ramification data. In recent years, several variants of this notion for genus $0$ base curves have appeared in the literature. Among them are so-called monotone Hurwitz numbers, which are related to the HCIZ integral in random matrix theory and strictly monotone Hurwitz numbers which count certain Grothendieck dessins d'enfants. We generalise the notion of Hurwitz numbers to interpolations between simple, monotone and strictly monotone Hurwitz numbers to any genus and any number of arbitrary but fixed ramification profiles. This yields generalisations of several results known for Hurwitz numbers. When the target surface is of genus one, we show that the generating series of these interpolated Hurwitz numbers are quasimodular forms. In the case that all ramification is simple, we refine this result by writing this series as a sum of quasimodular forms corresonding to tropical covers weighted by Gromov-Witten invariants. Moreover, we derive a quantum curve for monotone and Grothendieck dessins d'enfants Hurwitz numbers for arbitrary genera and one arbitrary but fixed ramification profile. Thus, we obtain spectral curves via the semiclassical limit as input data for the CEO topological recursion. Astonishingly, we find that the CEO topological recursion for the genus $1$ spectral curve of the strictly monotone Hurwitz numbers compute the monotone Hurwitz numbers in genus $0$. Thus, we give a new proof that monotone Hurwitz numbers satisfy CEO topological recursion. This points to an unknown relation between those enumerants. Finally, specializing to target surface $\mathbb{P}^1$, we find recursions for monotone and Grothendieck dessins d'enfants double Hurwitz numbers, which enables the computation of the respective Hurwitz numbers for any genera with one arbitrary but fixed ramification profile.Comment: 41 page

Wall-crossing formulae and strong piecewise polynomiality for mixed Grothendieck dessins d'enfant, monotone, and double simple Hurwitz numbers

We derive explicit formulae for the generating series of mixed Grothendieck dessins d'enfant/monotone/simple Hurwitz numbers, via the semi-infinite wedge formalism. This reveals the strong piecewise polynomiality in the sense of Goulden–Jackson–Vakil, generalising a result of Johnson, and provides a new explicit proof of the piecewise polynomiality of the mixed case. Moreover, we derive wall-crossing formulae for the mixed case. These statements specialise to any of the three types of Hurwitz numbers, and to the mixed case of any pair

Triply mixed coverings of arbitrary base curves : quasimodularity, quantum curves and a mysterious topological recursions

Simple Hurwitz numbers are classical invariants in enumerative geometry counting branched morphisms between Riemann surfaces with fixed ramification data. In recent years, several modifications of this notion for genus 0 base curves have appeared in the literature. Among them are so-called monotone Hurwitz numbers, which are related to the Harish–Chandra–Itzykson–Zuber integral in random matrix theory and strictly monotone Hurwitz numbers which enumerate certain Grothendieck dessins d’enfants. We generalise the notion of Hurwitz numbers to interpolations between simple, monotone and strictly monotone Hurwitz numbers for arbitrary genera and any number of arbitrary but fixed ramification profiles. This yields generalisations of several results known for Hurwitz numbers. When the target surface is of genus one, we show that the generating series of these interpolated Hurwitz numbers are quasimodular forms. In the case that all ramification is simple, we refine this result by writing this series as a sum of quasimodular forms corresponding to tropical covers weighted by Gromov–Witten invariants. Moreover, we derive a quantum curve for monotone and Grothendieck dessins d’enfants Hurwitz numbers for arbitrary genera and one arbitrary but fixed ramification profile. Thus, we obtain spectral curves via the semi-classical limit as input data for the Chekhov–Eynard–Orantin (CEO) topological recursion. Astonishingly, we find that the CEO topological recursion for the genus 1 spectral curve of the strictly monotone Hurwitz numbers computes the monotone Hurwitz numbers in genus 0. Thus, we give a new proof that monotone Hurwitz numbers satisfy CEO topological recursion. This points to an unknown relation between those enumerative invariants. Finally, specializing to target surface ℙ1, we find recursions for monotone and Grothendieck dessins d’enfants double Hurwitz numbers, which enables the computation of the respective Hurwitz numbers for any genera with one arbitrary but fixed ramification profile

Homological mirror symmetry for invertible polynomials in two variables

In this paper, we give a proof of homological mirror symmetry for two variable invertible polynomials, where the symmetry group on the B-side is taken to be maximal. The proof involves an explicit gluing construction of the Milnor fibres, and, as an application, we prove derived equivalences between certain nodal stacky curves, some of whose irreducible components have non-trivial generic stabiliser

Tropical Jucys covers

We study monotone and strictly monotone Hurwitz numbers from a bosonic Fock space perspective. This yields to an interpretation in terms of tropical geometry involving local multiplicities given by Gromov-Witten invariants. Furthermore, this enables us to prove that a main result of Cavalieri-Johnson-Markwig-Ranganathan is actually equivalent to the Gromov-Witten/Hurwitz correspondence by Okounkov-Pandharipande for the equivariant Riemann sphere

Towards a combinatorial classification of skew Schur functions

We present a single operation for constructing skew diagrams whose corresponding skew Schur functions are equal. This combinatorial operation naturally generalises and unifies all results of this type to date. Moreover, our operation suggests a closely related condition that we conjecture is necessary and sufficient for skew diagrams to yield equal skew Schur functions.Comment: 34 pages, 2 figures. Minor changes. Final version, to appear in Transactions of the AM