Representation-finiteness of endomorphism algebras for support ixed Gorenstein-projective modules

Determine whether, for every representation-finite string algebra A, the endomorphism algebra End_A(T) is representation-finite for every basic Gorenstein-projective support ixed-tilting A-module T.

Background

The paper proves that if a string algebra A satisfies the G-condition, then A is representation-finite if and only if End_A(T) is representation-finite for every basic Gorenstein-projective support ixed-tilting A-module T. Without the G-condition, the authors establish only partial results: representation-finiteness of the CohenMacaulay Auslander algebra implies representation-finiteness of the endomorphism algebras of all basic Gorenstein-projective modules, but a representation-finite string algebra can have a representation-infinite CohenMacaulay Auslander algebra.

The unresolved case concerns only basic Gorenstein-projective support ixed-tilting modules. The module G_A used to define the CohenMacaulay Auslander algebra need not itself be support ixed-tilting, so known counterexamples for the CohenMacaulay Auslander algebra do not resolve the stated question. Moreover, outside the G-condition, the relevant endomorphism algebras may be nonmonomial and therefore cannot be analyzed directly using the string-algebra band criterion.