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} ${\varphi }_A(T)$ 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 $\mathrm{mod}{}_A$ of finite dimensional left $A$-modules. We say that $A$ is of {\em cyclotomic type} if the characteristic polynomial ${\chi }_A$ of ${\varphi }_A$ 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 ${\mu }_0=\mathrm{1.176280...}$, 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} $A_m,\dots ,A_n$ 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 ${\rm Spec \,}ϕ_{A_n} \subset \s^1 \cup \R^+$ and $A_n$ is not of cyclotomic type then $M({\chi }_{A_m})<M({\chi }_{A_n})$.

arXiv admin note: substantial text overlap with arXiv:1310.1557