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Centrality of star and monotone factorisations

Jesse Campion Loth, Amarpreet Rattan

TL;DR

This work addresses the centrality of transitive star factorisations in $\mathfrak{S}_{n}$ by establishing a fully combinatorial proof valid in all genera. It shows the numbers of star, monotone, and transitive monotone factorisations are intricately connected via bijections, proving centrality through explicit mappings $\Gamma$ and $\Theta$ and a transitivity operator $T_n$ that renders $f(\Xi_n)$ central after transitive projection. The paper also uncovers new formulae for monotone double Hurwitz factorisations and ties these to geometry via branched covers and the HCIZ integral, while generalising centrality results for symmetric function evaluations of Jucys–Murphy elements. Overall, it provides a coherent combinatorial framework linking star and monotone factorisations, their centrality, and representations of the symmetric group, with several open questions about the transitive algebra generated by these constructions.

Abstract

A factorisation problem in the symmetric group is central if conjugate permutations always have the same number of factorisations. We give the first fully combinatorial proof of the centrality of transitive star factorisations that is valid in all genera, which answers a natural question of Goulden and Jackson from 2009. We begin by showing that the set of star factorisations is equinumerous with a certain set of monotone factorisations, a new result. We give more than one proof of this, and, crucially, one of our proofs is bijective. As a corollary we obtain new formulae for some monotone double Hurwitz factorisations, and a new relation between Hurwitz and monotone Hurwitz factorisations. We also generalise a theorem of Goulden and Jackson from 2009 that states that the transitive power of Jucys-Murphy elements are central. Our theorem states that the transitive image of any symmetric function evaluated at Jucys-Murphy elements is central, which gives a transitive version of Jucys' original result from 1974.

Centrality of star and monotone factorisations

TL;DR

This work addresses the centrality of transitive star factorisations in by establishing a fully combinatorial proof valid in all genera. It shows the numbers of star, monotone, and transitive monotone factorisations are intricately connected via bijections, proving centrality through explicit mappings and and a transitivity operator that renders central after transitive projection. The paper also uncovers new formulae for monotone double Hurwitz factorisations and ties these to geometry via branched covers and the HCIZ integral, while generalising centrality results for symmetric function evaluations of Jucys–Murphy elements. Overall, it provides a coherent combinatorial framework linking star and monotone factorisations, their centrality, and representations of the symmetric group, with several open questions about the transitive algebra generated by these constructions.

Abstract

A factorisation problem in the symmetric group is central if conjugate permutations always have the same number of factorisations. We give the first fully combinatorial proof of the centrality of transitive star factorisations that is valid in all genera, which answers a natural question of Goulden and Jackson from 2009. We begin by showing that the set of star factorisations is equinumerous with a certain set of monotone factorisations, a new result. We give more than one proof of this, and, crucially, one of our proofs is bijective. As a corollary we obtain new formulae for some monotone double Hurwitz factorisations, and a new relation between Hurwitz and monotone Hurwitz factorisations. We also generalise a theorem of Goulden and Jackson from 2009 that states that the transitive power of Jucys-Murphy elements are central. Our theorem states that the transitive image of any symmetric function evaluated at Jucys-Murphy elements is central, which gives a transitive version of Jucys' original result from 1974.
Paper Structure (12 sections, 11 theorems, 39 equations)

This paper contains 12 sections, 11 theorems, 39 equations.

Table of Contents

  1. Introduction
  2. Basics and notation
  3. Monotone factorisations
  4. Star factorisation and the transitivity operator
  5. Main results
  6. Connections to geometry and consequences of the main results
  7. Hurwitz moves
  8. A first proof of Theorem \ref{['thm:hurwitzrelation']}
  9. A combinatorial proof of centrality of monotone factorisations
  10. A bijection between star factorisations and monotone double Hurwitz factorisations
  11. The transitivity operator on symmetric polynomials of Jucys-Murphy elements
  12. Open problems

Key Result

Theorem 1

For any integers $k,n \geq 1$, Whence $e_k(\Upxi_n)$ is central.

Theorems & Definitions (17)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Corollary 5
  • Theorem 6
  • Theorem 7
  • proof
  • Corollary 8
  • proof
  • ...and 7 more