Cyclotomic factors of Coxeter polynomials

Benedict H. Gross, Eriko Hironaka, Curtis T. McMullen

In this paper we determine which roots of unity are zeros of the E_n Coxeter polynomial. We show that these roots come exclusively from splittings of E_n into spherical subdiagrams; in particular they have order 2, 3, 5, 8, 12, 18, or 30, and they only depend on the value of n mod 360 (provided we exclude the special case n = 9).