Opposite arrows obstruct finite representation type

Two distinct vertices joined by arrows in both directions form an oriented two-cycle. Such a quiver has infinite representation type over every field, including finite fields. This is the cycle obstruction that the underlying simple graph cannot detect: that graph records a two-cycle as a single edge.

Put the nilpotent Jordan block k[X]/(Xⁿ⁺¹) at both vertices, let the forward arrows act as the identity, and let the reverse arrows act by multiplication by X. The identity arrow forces an endomorphism to have equal components, and the reverse arrow forces that component to commute with X. Thus its idempotents are exactly those of the existing one-loop Jordan block. These representations are indecomposable and have distinct dimension vectors. Extension by zero then gives the obstruction in any quiver containing opposite arrows.

Main result

• TauCeti.not_isFiniteRepType_of_opposite_hom: opposite arrows between distinct vertices obstruct finite representation type.

References

• I. Assem, D. Simson, A. Skowroński, Elements of the Representation Theory of Associative Algebras, Volume I, Chapter VII.

The construction uses TauCeti.oneLoopNilpotentRep and its indecomposability theorem; the Jordan-block endomorphism calculation is reused rather than repeated.

theorem TauCeti.not_isFiniteRepType_of_opposite_hom {k : Type u} [Field k] {Q : Type v} [Quiver Q] {a b : Q} (hab : a ≠ b) (α : a ⟶ b) (β : b ⟶ a) : ¬IsFiniteRepType k Q

Opposite arrows between distinct vertices obstruct finite representation type over every field. No finiteness assumption on the ambient quiver is needed.