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

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.