Preprojective Hilbert series of non-Dynkin graphs as inverse zigzag Cartan matrices

Let G be a connected finite simple graph which is not a Dynkin diagram, o any orientation of G and H(t) the matrix Hilbert series of the preprojective algebra Π_k(Q) of the oriented quiver over a field k, indexed by the vertices of G. The oriented quiver carries a positive weight which is superadditive on its underlying multigraph (TauCeti.exists_pos_two_mul_le_sum_orientedQuiver_of_not_exists_dynkinType_iso, in Koszul/NonDynkin.lean), so by the theorem of Etingof and Eu the weighted inverse quantum Cartan comparison of HilbertSeries/ZigzagCartan.lean holds unconditionally:

H_{Π(Q)}(t) = C_G(-t)⁻¹,

where C_G(q) = (1 + q²) I + q A_G is the graded Cartan matrix of the zigzag algebra of G. This is an identity in the ring of matrices of formal power series over ℤ. It covers the non-Dynkin trees, such as D~ₙ, E₆~, E₇~ and E₈~, which have no orientation without sinks, together with every connected graph containing a cycle. For a finite simply-laced Dynkin diagram with at least two vertices the comparison fails: the preprojective algebra is finite-dimensional and C_G(-t)⁻¹ has a negative coefficient (TauCeti.exists_coeff_inv_zigzagGradedCartanMatrix_map_aeval_neg_X_lt_zero).

Main results

• C_G(-t) H(t) = I for every orientation of a connected non-Dynkin graph: zigzagGradedCartanMatrix_mul_preprojectiveHilbertSeries_eq_one_of_not_exists_dynkinType_iso.
• H_{Π(Q)}(t) = C_G(-t)⁻¹ for every orientation of a connected non-Dynkin graph: preprojectiveHilbertSeries_eq_inv_zigzagGradedCartanMatrix_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, for the Hilbert series of the preprojective algebra of a non-Dynkin quiver.
• R. S. Huerfano and M. Khovanov, A category for the adjoint representation, J. Algebra 246 (2001), for the graded Cartan matrix of the zigzag algebra.

theorem TauCeti.zigzagGradedCartanMatrix_mul_preprojectiveHilbertSeries_eq_one_of_not_exists_dynkinType_iso (k : Type w) [Field k] {V : Type u} {G : SimpleGraph V} [Fintype V] [DecidableEq V] (hG : G.Connected) (hD : ¬∃ (t : DynkinType), t.Valid ∧ Nonempty (G ≃g diagramGraph t.cartanMatrix)) (o : DoubledQuiver.Orientation G) : (zigzagGradedCartanMatrix k G).map ⇑(Polynomial.aeval (-PowerSeries.X)) * (preprojectiveHilbertSeries k (DoubledQuiver.OrientedQuiver G o)).submatrix (DoubledQuiver.OrientedQuiver.vertex G o) (DoubledQuiver.OrientedQuiver.vertex G o) = 1

The inverse quantum Cartan comparison for a connected non-Dynkin graph. For every orientation o of a connected finite simple graph G which is not a Dynkin diagram, the zigzag graded Cartan matrix C_G(q) = (1 + q²) I + q A_G evaluated at q = -X is a left inverse of the matrix Hilbert series of the preprojective algebra of the oriented quiver, indexed by the vertices of G.

theorem TauCeti.preprojectiveHilbertSeries_eq_inv_zigzagGradedCartanMatrix_of_not_exists_dynkinType_iso (k : Type w) [Field k] {V : Type u} {G : SimpleGraph V} [Fintype V] [DecidableEq V] (hG : G.Connected) (hD : ¬∃ (t : DynkinType), t.Valid ∧ Nonempty (G ≃g diagramGraph t.cartanMatrix)) (o : DoubledQuiver.Orientation G) : (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

The Hilbert series of the preprojective algebra of a connected non-Dynkin graph is the inverse zigzag Cartan matrix at q = -X. For every orientation o of a connected finite simple graph G which is not a Dynkin diagram, the matrix Hilbert series of the preprojective algebra of the oriented quiver over any field, indexed by the vertices of G, is H(X) = C_G(-X)⁻¹, where C_G(q) = (1 + q²) I + q A_G is the graded Cartan matrix of the zigzag algebra of G. This includes the non-Dynkin trees, whose orientations all have sinks.