On the Mahler measure of the Coxeter polynomials of algebras

TL;DR

This paper investigates the Mahler measure of Coxeter polynomials of accessible algebras, establishing bounds related to Lehmer's problem and introducing interlaced towers of algebras to analyze spectral properties and Mahler measures.

Contribution

It proves that for accessible algebras, the Mahler measure is either 1 or at least Lehmer's constant, and introduces interlaced towers of algebras to study spectral and Mahler measure growth.

Findings

Mahler measure of Coxeter polynomials is either 1 or exceeds Lehmer's constant.

Interlaced towers of algebras satisfy specific recurrence relations for their Coxeter polynomials.

Spectral conditions imply Mahler measure inequalities within algebra towers.

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 $M(\chi_{A_m}) <M(\chi_{A_n})$.