Extended Dynkin trees in the underlying graph obstruct finite representation type

The extended Dynkin quivers TauCeti.Quiver.AffineD m, TauCeti.Quiver.AffineE6, TauCeti.Quiver.AffineE7 and TauCeti.Quiver.AffineE8 have infinite representation type over every field, each in one fixed orientation. Their underlying graphs, the extended Dynkin diagrams D~ₘ₊₄, E₆~, E₇~ and E₈~, are trees. This file shows that each of them obstructs finite representation type wherever it occurs in the underlying graph of a quiver: if the underlying graph of Q contains a copy of one of these trees, not necessarily induced, then Q has infinite representation type, whatever the directions and multiplicities of the arrows of Q along the copy.

The deduction is TauCeti.IsFiniteRepType.of_copy_underlyingGraph: the arrows of Q along the copy orient the tree as a subquiver of Q, and finite representation type of a tree does not depend on its orientation.

These four trees are the extended Dynkin obstructions of Gabriel's theorem that are not cycles: a finite tree which is not a Dynkin diagram contains a copy of one of them.

Main results

• TauCeti.not_isFiniteRepType_of_copy_affineD: a quiver whose underlying graph contains D~ₘ₊₄ has infinite representation type.
• TauCeti.not_isFiniteRepType_of_copy_affineE6, TauCeti.not_isFiniteRepType_of_copy_affineE7 and TauCeti.not_isFiniteRepType_of_copy_affineE8: the same for E₆~, E₇~ and E₈~.

Implementation notes

The statements are for representations with vertex spaces in the universe of the base field, the universe in which the infinite families of the four extended Dynkin quivers are built.

References

• I. N. Bernstein, I. M. Gelfand, V. A. Ponomarev, Coxeter functors and Gabriel's theorem, Russian Math. Surveys 28 (1973), 17--32.
• I. Assem, D. Simson, A. Skowroński, Elements of the Representation Theory of Associative Algebras, Volume I, Chapter VII.

theorem TauCeti.not_isFiniteRepType_of_copy_affineD {k : Type u} [Field k] {Q : Type v} [Quiver Q] (m : ℕ) (f : (Quiver.underlyingGraph (Quiver.AffineD m)).Copy (Quiver.underlyingGraph Q)) : ¬IsFiniteRepType k Q

A copy of D~ₘ₊₄ in the underlying graph obstructs finite representation type. If the underlying graph of Q contains a copy of the extended Dynkin diagram D~ₘ₊₄, the underlying graph of TauCeti.Quiver.AffineD m, then Q has infinite representation type over every field.

theorem TauCeti.not_isFiniteRepType_of_copy_affineE6 {k : Type u} [Field k] {Q : Type v} [Quiver Q] (f : (Quiver.underlyingGraph Quiver.AffineE6).Copy (Quiver.underlyingGraph Q)) : ¬IsFiniteRepType k Q

A copy of E₆~ in the underlying graph obstructs finite representation type. If the underlying graph of Q contains a copy of the extended Dynkin diagram E₆~, the underlying graph of TauCeti.Quiver.AffineE6, then Q has infinite representation type over every field.

theorem TauCeti.not_isFiniteRepType_of_copy_affineE7 {k : Type u} [Field k] {Q : Type v} [Quiver Q] (f : (Quiver.underlyingGraph Quiver.AffineE7).Copy (Quiver.underlyingGraph Q)) : ¬IsFiniteRepType k Q

A copy of E₇~ in the underlying graph obstructs finite representation type. If the underlying graph of Q contains a copy of the extended Dynkin diagram E₇~, the underlying graph of TauCeti.Quiver.AffineE7, then Q has infinite representation type over every field.

theorem TauCeti.not_isFiniteRepType_of_copy_affineE8 {k : Type u} [Field k] {Q : Type v} [Quiver Q] (f : (Quiver.underlyingGraph Quiver.AffineE8).Copy (Quiver.underlyingGraph Q)) : ¬IsFiniteRepType k Q

A copy of E₈~ in the underlying graph obstructs finite representation type. If the underlying graph of Q contains a copy of the extended Dynkin diagram E₈~, the underlying graph of TauCeti.Quiver.AffineE8, then Q has infinite representation type over every field.