Chapter1 4 1. Cyclotomic and Littlewood Polynomials Associated to Algebras

Abstract 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 D b mod A of the module category mod A of finite dimensional left A -modules. In this paper we study the Mahler measure M χ A of the Coxeter polynomial χ A of certain algebras A . We consider in more detail two cases: (a) A is said to be cyclotomic if all eigenvalues of χ A are roots of unity; (b) A is said to be of Littlewood type if all coefficients of χ A are − 1 , 0 or 1 . We find criteria in order that A is of one of those types. In particular, we establish new records according to Mossingshoff’s list of Record Mahler measures of polynomials q with 1 < M q as small as possible, ordered by their number of roots outside the unit circle.