2013/10/06 by José-Antonio de la Peña, de la Peña, José-Antonio
Mathematics · #Advanced Combinatorial Mathematics #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Representation Theory (math.RT) #Rings and Algebras (math.RA)
paper · pdf · doi:10.48550/arxiv.1310.1910
openalex publication_date 2013/10/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 ϕA(T) as the automorphism of the Grothendieck group K0(A) induced by the Auslander-Reiten translation τ in the derived category \Derb(\modA) of the module category \modA of finite dimensional left A-modules. We say that A is of \em cyclotomic type if the characteristic polynomial χA of ϕA is a product of cyclotomic polynomials, equivalently, if the \em Mahler measure M(χA)=1. In \citePe 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 T10 + T9 - T7 - T6 - T5 - T4 - T3 + T + 1 has Mahler measure μ0 = 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(χA)=1 or M(χB) ≥ μ0 for some convex subcategory B of A. We introduce \em interlaced tower of algebras Am,…,An with m ≤ n-2 satisfying χ_As+1 =(T+1) χAs - T χ_As-1 for m+1 ≤ s ≤ n-1. We prove that, if \rm Spec ϕAn ⊂ \s1 ∪ \R+ and An is not of cyclotomic type then M(χAm)