Hilbert series of finite-dimensional preprojective algebras

Let Q be a finite quiver with at least one vertex, Π = Π_k(Q) its preprojective algebra over a field k, H its matrix Hilbert series (TauCeti.preprojectiveHilbertSeries) and A the arrow-count matrix of the doubled quiver, A_{v,w} = #(w ⟶ v). Write M = (1 + X²) I - X A, a matrix of formal power series which is invertible because its constant term is I. When Π is finite-dimensional, its graded corners e_j Π_n e_i vanish in all large degrees, so the entries of H are polynomials. This file proves that then

• M H ≠ I, so the Koszul complex of some vertex module is not exact at its left end: some nonzero y ∈ e_v Π is killed by the reverse b* of every arrow b of the doubled quiver into v;
• the inverse M⁻¹ has a negative coefficient.

Both statements rest on one observation: when Q is nonempty, no matrix N whose entries are polynomials satisfies M N = I (Matrix.isEmpty_of_one_add_X_sq_smul_sub_X_smul_mul_eq_one). For the second statement, the Koszul defect formula gives M H = I + D where D has nonnegative coefficients (TauCeti.coeff_one_le_coeff_mul_preprojectiveHilbertSeries). If every coefficient of M⁻¹ were nonnegative, then H = M⁻¹ + M⁻¹ D would bound M⁻¹ coefficientwise, so M⁻¹ would be a matrix of polynomials.

For a finite simple graph G without isolated vertices, M is the zigzag graded Cartan matrix C_G(q) = (1 + q²) I + q A_G at q = -X (TauCeti.zigzagGradedCartanMatrix_map_aeval_neg_X_eq_submatrix). Hence C_G(-X)⁻¹ has a negative coefficient as soon as the preprojective algebra of one orientation of G is finite-dimensional, which is the case for every finite simply-laced Dynkin diagram (TauCeti.finiteDimensional_preprojectiveAlgebra_of_iso). This is the numerical obstruction to the Koszulity of the zigzag algebras of these graphs: the inverse of the graded Cartan matrix of a Koszul algebra at q = -X is the matrix Hilbert series of its Koszul dual, so it has nonnegative coefficients. It contrasts with the finite preconnected graphs containing a cycle, for which H = C_G(-X)⁻¹ (TauCeti.RepresentationTheory.Quiver.Preprojective.HilbertSeries.NonTree).

Main results

• TauCeti.exists_forall_le_preprojectiveCorner_eq_bot: the graded corners of a finite-dimensional preprojective algebra vanish in all large degrees.
• TauCeti.mul_preprojectiveHilbertSeries_ne_one: for a finite-dimensional preprojective algebra of a nonempty quiver, M H ≠ I.
• TauCeti.exists_ne_zero_forall_preprojectiveMk_ofArrow_reverse_mul_eq_zero: the Koszul complex of some vertex module is not exact at its left end.
• TauCeti.exists_coeff_inv_lt_zero_of_finiteDimensional: M⁻¹ has a negative coefficient.
• TauCeti.exists_coeff_inv_zigzagGradedCartanMatrix_map_aeval_neg_X_lt_zero: C_G(-X)⁻¹ has a negative coefficient when the preprojective algebra of an orientation of G is finite-dimensional.

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 Koszul complex and the Hilbert series.
• A. Beilinson, V. Ginzburg and W. Soergel, Koszul duality patterns in representation theory, J. Amer. Math. Soc. 9 (1996), Lemma 2.11.1, for the Hilbert series of a Koszul algebra and its dual.
• S. Brenner, M. C. R. Butler and A. D. King, Periodic algebras which are almost Koszul, Algebr. Represent. Theory 5 (2002), for the preprojective algebras of Dynkin quivers as almost Koszul algebras.

theorem TauCeti.exists_forall_le_preprojectiveCorner_eq_bot (k : Type w) (Q : Type u) [Field k] [Quiver Q] [Fintype Q] [(i j : Q) → Fintype (i ⟶ j)] [FiniteDimensional k (preprojectiveAlgebra k Q)] : ∃ (d : ℕ), ∀ (n : ℕ), d ≤ n → ∀ (i j : Quiver.Symmetrify Q), preprojectiveCorner k Q n i j = ⊥

The graded corners of a finite-dimensional preprojective algebra vanish in large degrees: there is a degree d with e_j Π_n e_i = 0 for all n ≥ d and all vertices i and j.

theorem TauCeti.mul_preprojectiveHilbertSeries_ne_one (k : Type w) (Q : Type u) [Field k] [Quiver Q] [Fintype Q] [(i j : Q) → Fintype (i ⟶ j)] [DecidableEq Q] [Nonempty Q] [FiniteDimensional k (preprojectiveAlgebra k Q)] : ((1 + PowerSeries.X ^ 2) • 1 - PowerSeries.X • Matrix.of fun (v w : Q) => ↑(Fintype.card (Quiver.Symmetrify.of.obj w ⟶ Quiver.Symmetrify.of.obj v))) * preprojectiveHilbertSeries k Q ≠ 1

The Hilbert series identity fails for a finite-dimensional preprojective algebra. If Q has a vertex, A_{v,w} = #(w ⟶ v) is the arrow-count matrix of the doubled quiver and Π is finite-dimensional, then its matrix Hilbert series H satisfies ((1 + X²) I - X A) H ≠ I.

theorem TauCeti.exists_ne_zero_forall_preprojectiveMk_ofArrow_reverse_mul_eq_zero (k : Type w) (Q : Type u) [Field k] [Quiver Q] [Fintype Q] [(i j : Q) → Fintype (i ⟶ j)] [Nonempty Q] [FiniteDimensional k (preprojectiveAlgebra k Q)] : ∃ (v : Q) (y : preprojectiveAlgebra k Q), (preprojectiveMk k Q) (doubledVertexIdempotent k v) * y = y ∧ (∀ (w : Quiver.Symmetrify Q) (b : w ⟶ Quiver.Symmetrify.of.obj v), (preprojectiveMk k Q) (PathAlgebra.ofArrow (Quiver.reverse b)) * y = 0) ∧ y ≠ 0

The Koszul complex of a finite-dimensional preprojective algebra is not exact at its left end. If Q has a vertex and Π is finite-dimensional, then at some vertex v there is a nonzero y ∈ e_v Π killed by the reverse b* of every arrow b of the doubled quiver into v: the left-hand map y ↦ (ε_b b* y)_b of the Koszul complex of the vertex module S_v is not injective.

theorem TauCeti.exists_coeff_inv_lt_zero_of_finiteDimensional (k : Type w) (Q : Type u) [Field k] [Quiver Q] [Fintype Q] [(i j : Q) → Fintype (i ⟶ j)] [DecidableEq Q] [Nonempty Q] [FiniteDimensional k (preprojectiveAlgebra k Q)] : ∃ (n : ℕ) (v : Q) (i : Q), (PowerSeries.coeff n) (((1 + PowerSeries.X ^ 2) • 1 - PowerSeries.X • Matrix.of fun (v w : Q) => ↑(Fintype.card (Quiver.Symmetrify.of.obj w ⟶ Quiver.Symmetrify.of.obj v)))⁻¹ v i) < 0

The inverse of (1 + X²) I - X A has a negative coefficient when Q has a vertex and its preprojective algebra is finite-dimensional, for A_{v,w} = #(w ⟶ v) the arrow-count matrix of the doubled quiver. The inverse is taken in the ring of matrices of formal power series over ℤ.

theorem TauCeti.exists_coeff_inv_zigzagGradedCartanMatrix_map_aeval_neg_X_lt_zero (k : Type w) [Field k] {V : Type u} {G : SimpleGraph V} [Fintype V] [DecidableEq V] [Nonempty V] (hns : ∀ (i : V), ∃ (j : V), G.Adj i j) (o : DoubledQuiver.Orientation G) [FiniteDimensional k (preprojectiveAlgebra k (DoubledQuiver.OrientedQuiver G o))] : ∃ (n : ℕ) (v : V) (w : V), (PowerSeries.coeff n) (((zigzagGradedCartanMatrix k G).map ⇑(Polynomial.aeval (-PowerSeries.X)))⁻¹ v w) < 0

The inverse zigzag Cartan matrix at q = -X has a negative coefficient for a graph with a finite-dimensional preprojective algebra. Let G be a nonempty finite simple graph without isolated vertices, and let the preprojective algebra of some orientation o of G be finite-dimensional. Then the inverse of the zigzag graded Cartan matrix C_G(q) = (1 + q²) I + q A_G evaluated at q = -X, taken in the ring of matrices of formal power series over ℤ, has a negative coefficient.