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Spectral Growth in $W(E_{10})$: Double Coset Filtration and Hilbert Geometry

Kyounghee Kim

TL;DR

This work introduces an $s_0$-level filtration of the hyperbolic Coxeter group $W(E_{10})$ using $W(A_9)$-double cosets and a triple-based combinatorial model to encode minimal double coset representatives. It couples this algebraic structure with a geometric Hilbert-geometry framework on the Tits cone, linking spectral radii to Hilbert translation lengths and providing a recursive, compute-friendly method for enumerating spectral data. The construction yields a rooted DAG of triples, a Demazure-type monoid on minimal representatives, and a parabolic Hecke-algebra viewpoint that together illuminate the Weyl spectrum of $W(E_{10})$ and its level-by-level growth. The approach generalizes naturally to $W(E_n)$ and connects to dynamics on rational surfaces, offering a geometric and combinatorial lens on entropy spectra of surface automorphisms.

Abstract

We study the spectral radii of elements in the hyperbolic Coxeter group $W(E_{10})$ by introducing a filtration indexed by reflections conjugate to a distinguished simple reflection $s_0$. This filtration organizes $W(E_{10})$ into double cosets relative to the parabolic subgroup $W(A_9)$, and we classify the minimal representatives of these cosets via a rooted directed acyclic graph (DAG) labeled by triples. Each node in the DAG corresponds to a structured reflection composition, enabling a recursive understanding of spectral growth. Using the Hilbert metric on the Tits cone, we relate spectral radii to geometric displacement and demonstrate an effective method to compute the spectral radii inductively. This provides a geometric and combinatorial framework for understanding the Weyl spectrum of $W(E_{10})$. While our focus is on $E_{10}$, the techniques developed extended naturally to the family $W(E_n)$ for $n\ge 10$, with implications for dynamics on rational surfaces and entropy spectra of surface automorphisms.

Spectral Growth in $W(E_{10})$: Double Coset Filtration and Hilbert Geometry

TL;DR

This work introduces an -level filtration of the hyperbolic Coxeter group using -double cosets and a triple-based combinatorial model to encode minimal double coset representatives. It couples this algebraic structure with a geometric Hilbert-geometry framework on the Tits cone, linking spectral radii to Hilbert translation lengths and providing a recursive, compute-friendly method for enumerating spectral data. The construction yields a rooted DAG of triples, a Demazure-type monoid on minimal representatives, and a parabolic Hecke-algebra viewpoint that together illuminate the Weyl spectrum of and its level-by-level growth. The approach generalizes naturally to and connects to dynamics on rational surfaces, offering a geometric and combinatorial lens on entropy spectra of surface automorphisms.

Abstract

We study the spectral radii of elements in the hyperbolic Coxeter group by introducing a filtration indexed by reflections conjugate to a distinguished simple reflection . This filtration organizes into double cosets relative to the parabolic subgroup , and we classify the minimal representatives of these cosets via a rooted directed acyclic graph (DAG) labeled by triples. Each node in the DAG corresponds to a structured reflection composition, enabling a recursive understanding of spectral growth. Using the Hilbert metric on the Tits cone, we relate spectral radii to geometric displacement and demonstrate an effective method to compute the spectral radii inductively. This provides a geometric and combinatorial framework for understanding the Weyl spectrum of . While our focus is on , the techniques developed extended naturally to the family for , with implications for dynamics on rational surfaces and entropy spectra of surface automorphisms.

Paper Structure

This paper contains 20 sections, 25 theorems, 155 equations, 3 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Coxeter group $W(E_{10})$
  3. Bruhat Order
  4. $W(A_9)$-double cosets
  5. Representatives of Double Cosets
  6. Representing Triples
  7. Recursive construction of $T_n$ from $T_2$
  8. Adjacency filter.
  9. Level test via the three--$\kappa$ lemma:
  10. Double--coset identification via matrices:
  11. Algebraic structure of the double coset
  12. The $s_0$-Level Filtration of $W(E_{10})$
  13. Spectral refinement and growth graph of minimal representatives
  14. Spectral Radii via the Hilbert Metric
  15. Hilbert metric.
  16. ...and 5 more sections

Key Result

Theorem 1

Let $\mathcal{D}_{n,n}$ denote the set of elements in $W(E_{10})$ of $s_0$--level $n$ (as defined in E:Dnn), and let $K \subset W(E_{10})$ be the set of Bruhat-minimal representatives of double cosets $W(A_9)\backslash W(E_{10})/W(A_9)$. Then:

Figures (3)

  • Figure 1: Dynkin diagram of $E_{10}$
  • Figure 2: Branching graph for $T_1/\!\sim \to T_2/\!\sim \to T_3/\!\sim$. Each directed edge represents the extension process in which a new ordered set of triples $\widehat{I}$ is obtained by adjoining the triple indicated at the end of the branch as the first triple in the sequence. For clarity, only the leftmost directed paths are shown, so that each node is labeled by the triples appearing along that path.
  • Figure 3: Exponential decay fit to the average gaps in $\mathrm{Gap}_n$ between newly emerged Salem numbers in $\tilde{\Lambda}_n^\text{new}$. Purple dots show observed means; the curve is a best-fit exponential regression obtained using Mathematica.

Theorems & Definitions (44)

  • Theorem : Triple-Labeled Graph Structure of Minimal Representatives, Theorem \ref{['T:TripleGraph']}
  • Theorem : Spectral Monotonicity, Theorem \ref{['T:monotonicity']}
  • Theorem : Spectral Exhaustion Theorem, Theorem \ref{['T:exhaustion']}
  • Theorem : Hilbert Ball Inclusion of Level-$n$ Elements, Theorem \ref{['T:hilbertBall']}
  • Conjecture : Spectral Growth Conjecture
  • Remark 1.1
  • Theorem 2.1: Matsumoto's Theorem BjornerBrenti
  • Lemma 2.2: BjornerBrenti
  • Lemma 3.1
  • proof
  • ...and 34 more