On the Mahler measure of the Coxeter polynomial of an algebra

Let A be a finite dimensional algebra over an algebraically closed field k. Assume A is a basic connected and triangular algebra with n pairwise non-isomorphic simple modules. We consider the Coxeter transformation ϕ A ( T ) as the automorphism of the Grothendieck group K 0 ( A ) induced by the Auslander–Reiten translation τ in the derived category Der b ( mod A ) of the module category mod A of finite dimensional left A-modules. We say that A is of cyclotomic type if the characteristic polynomial χ A of ϕ A is a product of cyclotomic polynomials, equivalently, if the Mahler measure M ( χ A ) = 1 . In [6] we have considered many examples of algebras of cyclotomic type in the representation theory literature. In this paper we study the Mahler measure of the Coxeter polynomial of accessible algebras. In 1933, D.H. Lehmer found that the polynomial T 10 + T 9 − T 7 − T 6 − T 5 − T 4 − T 3 + T + 1 has Mahler measure μ 0 = 1.176280 . . . , and he asked whether there exist any smaller values exceeding 1. In this paper we prove that for any accessible algebra A either M ( χ A ) = 1 or M ( χ B ) ≥ μ 0 for some convex subcategory B of A. We introduce interlaced tower of algebras A m , … , A n with m ≤ n − 2 satisfying χ A s + 1 = ( T + 1 ) χ A s − T χ A s − 1 for m + 1 ≤ s ≤ n − 1 . We prove that, if Spec ϕ A n ⊂ S 1 ∪ R + and A n is not of cyclotomic type, then M ( χ A m ) M ( χ A n ) . We construct several examples.