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Representation asymptotics in the homology of pure graph braid groups

Louis Hainaut, Ben Knudsen, Nicholas Wawrykow

TL;DR

The paper develops an explicit asymptotic description of the homology of ordered configuration spaces $F_k(\\mathsf{\\Gamma})$ for graphs, giving precise growth rates and representation-theoretic multiplicities over arbitrary fields. Central to the method is a twisted-algebra framework and the vertex-explosion spectral sequence, whose leading $p=0$ terms are controlled by disjoint unions of star subgraphs, while higher $p$-terms are bounded via a cascade of Tor calculations in a Koszul-like complex. In characteristic zero, the authors provide detailed formulas for multiplicities of many Specht modules and relate their growth to graph invariants $\\Lambda^i_{\\mathsf{\\Gamma}}$, $\\Delta^i_{\\mathsf{\\Gamma}}$, and $\\mathcal{E}_{\\mathsf{\\Gamma}}^i$. The results yield asymptotically universal generators and confirm stability phenomena in a non-Noetherian setting, opening new avenues for precise asymptotics in graph braid groups and their representation theory. Overall, the work significantly advances the understanding of how graph topology governs the asymptotic algebraic structure of configuration-space homology.

Abstract

We give explicit formulas for the asymptotic Betti numbers, over an arbitrary field, of the ordered configuration spaces of a graph. In characteristic zero, we further give explicit formulas for the asymptotic multiplicities in homology of many irreducible representations of the symmetric group, in the spirit of representation stability.

Representation asymptotics in the homology of pure graph braid groups

TL;DR

The paper develops an explicit asymptotic description of the homology of ordered configuration spaces for graphs, giving precise growth rates and representation-theoretic multiplicities over arbitrary fields. Central to the method is a twisted-algebra framework and the vertex-explosion spectral sequence, whose leading terms are controlled by disjoint unions of star subgraphs, while higher -terms are bounded via a cascade of Tor calculations in a Koszul-like complex. In characteristic zero, the authors provide detailed formulas for multiplicities of many Specht modules and relate their growth to graph invariants , , and . The results yield asymptotically universal generators and confirm stability phenomena in a non-Noetherian setting, opening new avenues for precise asymptotics in graph braid groups and their representation theory. Overall, the work significantly advances the understanding of how graph topology governs the asymptotic algebraic structure of configuration-space homology.

Abstract

We give explicit formulas for the asymptotic Betti numbers, over an arbitrary field, of the ordered configuration spaces of a graph. In characteristic zero, we further give explicit formulas for the asymptotic multiplicities in homology of many irreducible representations of the symmetric group, in the spirit of representation stability.

Paper Structure

This paper contains 24 sections, 38 theorems, 76 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Torsion
  3. Universal generators
  4. Stability phenomena
  5. Outline of argument
  6. Conventions
  7. Acknowledgements
  8. Twisted algebra
  9. Basic notions
  10. Some results on Tor
  11. Configuration spaces of graphs
  12. Graphs and leaf stabilization
  13. Świątkowski complex
  14. Asymptotics of functions and symmetric sequences
  15. Functions of integers
  16. ...and 9 more sections

Key Result

Theorem 1.1

If $\mathsf{\Gamma}$ is a connected graph not homeomorphic to the circle such that $\Lambda^i_\mathsf{\Gamma}>0$, then we have where $S$ ranges over sets of essential vertices of cardinality $i$, and $\approx$ denotes equality modulo $o(k^{\Lambda^i_\mathsf{\Gamma}-1}k!)$. In characteristic zero, the same conclusion holds after substituting $\Delta^i_\mathsf{\Gamma}>1$, $\mathrm{mul}_\lambda$, an

Figures (3)

  • Figure 1: The star graphs $\mathsf{S}_{3}$, $\mathsf{S}_{4}$, and $\mathsf{S}_{5}$.
  • Figure 2: A graph and its explosion at a set of three bivalent vertices.
  • Figure 3: A configuration of four particles and its image under leaf stabilization.

Theorems & Definitions (88)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Remark 1.4
  • Remark 1.5
  • Remark 1.6
  • Conjecture 1.7: Folklore
  • Corollary 1.8
  • Conjecture 1.9
  • Corollary 1.10
  • ...and 78 more