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Given a finite dimensional algebra of finite global dimension. In https://arxiv.org/pdf/0805.1018.pdf it is mentioned that for some classes of such algebras (for example quiver algebras) the Coxeter polynomial has spectral radius one. This implies that the Coxeter polynomial of the algebra is a product of cyclotomic polynomials.

Questions:

  1. Which products of cyclotomic polynomials can be realised as Coxeter polynomial of a (quiver) algebra of finite global dimension?

  2. Are there applications to the algebra, when one knows the decomposition of the Coxeter polynomial into cyclotomic polynomials (or other applications knowing that the decomposition exists)?

  3. Is there an easy way to check, whether the Coxeter polynomial has this property?

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    $\begingroup$ Have you seen the paper of José Antonio de la Peña: "Algebras whose Coxeter polynomials are products of cyclotomic polynomials." Algebr. Represent. Theory 17 (2014), no. 3, 905–930? $\endgroup$
    – Alex Dugas
    Commented Mar 7, 2017 at 22:31

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