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For a hereditary, finite-dimensional algebra $A$ the Coxeter transformationextends the action of the Auslander--Reiten translation on the non-projectiveindecomposable modules to a linear endomorphism of the Grothendieck group ofthe category of finitely generated $A$-modules. It is natural to ask whetherother algebras admit a similar linear extension. We show that this is indeedthe case for all Nakayama algebras. Conversely, we show that finite-dimensionalalgebras with non-acyclic and connected quiver admitting such a linearextension are already cyclic Nakayama algebras.

When does the Auslander-Reiten translation operate linearly on the Grothendieck group? -- Part I