On the Mahler measure of the Coxeter polynomials of 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 {\em Coxeter transformation} $\phi_A(T)$ as the automorphism of the Grothendieck group $K_0(A)$ induced by the Auslander-Reiten translation $\tau$ 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 {\em cyclotomic type} if the characteristic polynomial $\chi_A$ of $\phi_A$ is a product of cyclotomic polynomials, equivalently, if the {\em Mahler measure} $M(\chi_A)=1$. In \cite{Pe} we have considered the 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 {\em 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 $\mu_0 = 1.176280 . . .$, and he asked if there exist any smaller values exceeding 1. In this paper we prove that for any accessible algebra $A$ either $M(\chi_A)=1$ or $M(\chi_B) \ge \mu_0$ for some convex subcategory $B$ of $A$. We introduce {\em interlaced tower of algebras} $A_m,\ldots,A_n$ with $m \le n-2$ satisfying $$\chi_{A_{s+1}} =(T+1) \chi_{A_s} - T \chi_{A_{s-1}}$$ for $m+1 \le s \le n-1$. We prove that, if ${\rm Spec \,}\phi_{A_n} \subset \s^1 \cup \R^+$ and $A_n$ is not of cyclotomic type then $M(\chi_{A_m}) <M(\chi_{A_n})$.