Hilbert series for every orientation of a graph containing a cycle

For a finite preconnected simple graph containing a cycle, every orientation has exact vertex Koszul complexes and linear graded projective resolutions. Consequently its preprojective matrix Hilbert series is the inverse of (1 + X²) I - X A, where A is the adjacency matrix. The chosen orientation can have sinks: these statements use orientation-independent injectivity from TauCeti.RepresentationTheory.Quiver.Preprojective.NonTree.

The homogeneous corner dimensions satisfy the resulting second-order recursion. Reindexing by the vertices of the graph identifies the denominator with the zigzag graded Cartan matrix at q = -X. This is an identity of formal power series, with no claim about a finite Ext-Euler pairing. Non-Dynkin trees are not covered here; over a field the Hilbert series of every connected non-Dynkin graph is TauCeti.preprojectiveHilbertSeries_eq_inv_of_not_exists_dynkinType_iso.

References

• P. Etingof and C.-H. Eu, Koszulity and the Hilbert series of preprojective algebras, Math. Res. Lett. 14 (2007), Sections 2 and 3.
• R. S. Huerfano and M. Khovanov, A category for the adjoint representation, J. Algebra 246 (2001), for the zigzag graded Cartan matrix.

theorem TauCeti.sum_card_mul_finrank_preprojectiveCorner_eq_of_not_isAcyclic (k : Type w) [Field k] {V : Type u} {G : SimpleGraph V} (hG : G.Preconnected) (hc : ¬G.IsAcyclic) (o : DoubledQuiver.Orientation G) [Finite V] (m : ℕ) (i : Quiver.Symmetrify (DoubledQuiver.OrientedQuiver G o)) (v : DoubledQuiver.OrientedQuiver G o) : ∑ w : Quiver.Symmetrify (DoubledQuiver.OrientedQuiver G o), Fintype.card (w ⟶ Quiver.Symmetrify.of.obj v) * Module.finrank k ↥(preprojectiveCorner k (DoubledQuiver.OrientedQuiver G o) (m + 1) i w) = Module.finrank k ↥(preprojectiveCorner k (DoubledQuiver.OrientedQuiver G o) (m + 2) i (Quiver.Symmetrify.of.obj v)) + Module.finrank k ↥(preprojectiveCorner k (DoubledQuiver.OrientedQuiver G o) m i (Quiver.Symmetrify.of.obj v))

The homogeneous corner dimensions of every orientation of a finite preconnected graph containing a cycle obey the preprojective Hilbert-series recursion. The middle sum counts arrows of the doubled quiver, independently of the direction of the original arrows.

theorem TauCeti.mul_preprojectiveHilbertSeries_eq_one_of_not_isAcyclic (k : Type w) [Field k] {V : Type u} {G : SimpleGraph V} (hG : G.Preconnected) (hc : ¬G.IsAcyclic) (o : DoubledQuiver.Orientation G) [Finite V] [DecidableEq (DoubledQuiver.OrientedQuiver G o)] : ((1 + PowerSeries.X ^ 2) • 1 - PowerSeries.X • Matrix.of fun (v w : DoubledQuiver.OrientedQuiver G o) => ↑(Fintype.card (Quiver.Symmetrify.of.obj w ⟶ Quiver.Symmetrify.of.obj v))) * preprojectiveHilbertSeries k (DoubledQuiver.OrientedQuiver G o) = 1

For every orientation of a finite preconnected graph containing a cycle, ((1 + X²) I - X A) H = I, where A counts doubled arrows and H is the matrix Hilbert series of the preprojective algebra.

theorem TauCeti.preprojectiveHilbertSeries_eq_inv_of_not_isAcyclic (k : Type w) [Field k] {V : Type u} {G : SimpleGraph V} (hG : G.Preconnected) (hc : ¬G.IsAcyclic) (o : DoubledQuiver.Orientation G) [Finite V] [DecidableEq (DoubledQuiver.OrientedQuiver G o)] : preprojectiveHilbertSeries k (DoubledQuiver.OrientedQuiver G o) = ((1 + PowerSeries.X ^ 2) • 1 - PowerSeries.X • Matrix.of fun (v w : DoubledQuiver.OrientedQuiver G o) => ↑(Fintype.card (Quiver.Symmetrify.of.obj w ⟶ Quiver.Symmetrify.of.obj v)))⁻¹

The preprojective matrix Hilbert series of every orientation of a finite preconnected graph containing a cycle is ((1 + X²) I - X A)⁻¹. This formula holds over every field, including characteristic two.

theorem TauCeti.preprojectiveHilbertSeries_eq_inv_zigzagGradedCartanMatrix_map_aeval_neg_X_of_not_isAcyclic (k : Type w) [Field k] {V : Type u} {G : SimpleGraph V} (hG : G.Preconnected) (hc : ¬G.IsAcyclic) (o : DoubledQuiver.Orientation G) [Fintype V] [DecidableEq V] : (preprojectiveHilbertSeries k (DoubledQuiver.OrientedQuiver G o)).submatrix (DoubledQuiver.OrientedQuiver.vertex G o) (DoubledQuiver.OrientedQuiver.vertex G o) = ((zigzagGradedCartanMatrix k G).map ⇑(Polynomial.aeval (-PowerSeries.X)))⁻¹

For every orientation of a finite preconnected graph containing a cycle, the matrix Hilbert series of its preprojective algebra, indexed by graph vertices, is the inverse zigzag graded Cartan matrix at q = -X. The orientation may have sinks.