Preprojective Hilbert series as inverse zigzag Cartan matrices #
Let Π = Π_k(Q) be the preprojective algebra of a finite quiver Q over a field k, with matrix
Hilbert series H(t) (TauCeti.preprojectiveHilbertSeries), and let A be the arrow-count matrix
of the doubled quiver. By TauCeti.mul_preprojectiveHilbertSeries_eq_one_iff, the identity
((1 + t²) I - t A) H(t) = I holds exactly when the Koszul complexes of the vertex modules are
projective resolutions. When Q has no sinks, they are
(TauCeti.forall_preprojectiveMk_ofArrow_reverse_mul_eq_zero_iff), so
H(t) = ((1 + t²) I - t A)⁻¹
in the ring of matrices of formal power series, and the dimensions of the graded corners obey the
recursion dim e_v Π_{m+2} e_i + dim e_v Π_m e_i = ∑_w #(w ⟶ v) dim e_w Π_{m+1} e_i.
When Q is an orientation of a finite simple graph G without isolated vertices, A is the
adjacency matrix of G, so (1 + t²) I - t A is the graded Cartan matrix
C_G(q) = (1 + q²) I + q A_G of the zigzag algebra of G (TauCeti.zigzagGradedCartanMatrix)
evaluated at q = -t. This compares the inverse quantum Cartan matrix of the zigzag algebra, in the
power-series completion, with the graded dimensions of the preprojective algebra, which is its
quadratic dual for bipartite G:
H_{Π(Q)}(t) = C_G(-t)⁻¹.
It holds for every orientation of G at which every Koszul complex is a resolution. This file
proves the identity unconditionally as soon as the oriented quiver carries a positive weight δ
satisfying 2 δ_i ≤ ∑_j (#(i ⟶ j) + #(j ⟶ i)) δ_j at every vertex of G. Orientations without
sinks are covered separately
(TauCeti.preprojectiveHilbertSeries_eq_inv_zigzagGradedCartanMatrix_map_aeval_neg_X); a
connected graph which is not a Dynkin diagram carries such a weight in every orientation
(TauCeti.exists_pos_two_mul_le_sum_orientedQuiver_of_not_exists_dynkinType_iso), so by the
theorem of Etingof and Eu the identity holds for every orientation
(TauCeti.preprojectiveHilbertSeries_eq_inv_zigzagGradedCartanMatrix_of_not_exists_dynkinType_iso,
module HilbertSeries/NonDynkin.lean), covering the non-Dynkin trees, such as D~ₙ and E₆~,
E₇~, E₈~.
Main results #
TauCeti.sum_card_mul_finrank_preprojectiveCorner_eq: for a quiver without sinks, the dimension recursion of the graded corners holds in every degree.TauCeti.mul_preprojectiveHilbertSeries_eq_oneandTauCeti.preprojectiveHilbertSeries_eq_inv: for a quiver without sinks,H(t) = ((1 + t²) I - t A)⁻¹.TauCeti.zigzagGradedCartanMatrix_map_aeval_neg_X_mul_preprojectiveHilbertSeries_eq_one_iff: for an orientation of a graph without isolated vertices,C_G(-t) H(t) = Iexactly when every Koszul complex of the preprojective algebra is a resolution.zigzagGradedCartanMatrix_map_aeval_neg_X_mul_preprojectiveHilbertSeries_eq_one_of_two_mul_le_sumandpreprojectiveHilbertSeries_eq_inv_zigzagGradedCartanMatrix_map_aeval_neg_X_of_two_mul_le_sum(both namespaced inTauCeti):H_{Π(Q)}(t) = C_G(-t)⁻¹for an orientation whose oriented quiver carries a positive superadditive weight.TauCeti.zigzagGradedCartanMatrix_map_aeval_neg_X_mul_preprojectiveHilbertSeries_eq_oneandTauCeti.preprojectiveHilbertSeries_eq_inv_zigzagGradedCartanMatrix_map_aeval_neg_X: for an orientation without sinks,H_{Π(Q)}(t) = C_G(-t)⁻¹.
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
H(t) = ((1 + t²) I - t A)⁻¹of 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 and its quadratic dual.
The dimension recursion for a quiver without sinks. If every vertex of Q is the tail of an
arrow, then for all vertices i and v and every degree m,
∑_w #(w ⟶ v) dim e_w Π_{m+1} e_i = dim e_v Π_{m+2} e_i + dim e_v Π_m e_i,
the sum running over the vertices w of the doubled quiver.
The Hilbert series identity for a quiver without sinks. If every vertex of Q is the tail
of an arrow and A_{v,w} = #(w ⟶ v) is the arrow-count matrix of the doubled quiver, then
((1 + X²) I - X A) H = I for the matrix Hilbert series H of Π.
The Hilbert series of the preprojective algebra of a quiver without sinks is the inverse
matrix H = ((1 + X²) I - X A)⁻¹, for A_{v,w} = #(w ⟶ v) the arrow-count matrix of the doubled
quiver.
Over a graph without isolated vertices, the zigzag graded Cartan matrix at q = -X is the
denominator of the preprojective Hilbert series of any orientation, read through the
identification of the vertices of the oriented quiver with those of G.
The inverse quantum Cartan comparison, conditionally. Let G be a finite simple graph
without isolated vertices, o an orientation of G, and H the matrix Hilbert series of the
preprojective algebra of the oriented quiver, indexed by the vertices of G. Then the zigzag graded
Cartan matrix C_G(q) = (1 + q²) I + q A_G evaluated at q = -X satisfies C_G(-X) H = I exactly
when at every vertex v the only y ∈ e_v Π killed by the reverse b* of every arrow b of the
doubled quiver into v is 0, that is, exactly when every Koszul complex of Π is a projective
resolution.
The inverse quantum Cartan comparison with a superadditive weight. Let G be a finite
simple graph, o an orientation of G, and suppose the oriented quiver carries a positive weight
δ over a strictly ordered commutative ring with 2 δ i ≤ ∑_j (#(i ⟶ j) + #(j ⟶ i)) δ_j at
every vertex. Then C_G(-X) H = I, where H is the matrix Hilbert series of the preprojective
algebra of the oriented quiver, indexed by the vertices of G. The non-Dynkin hypothesis of the
theorem of Etingof and Eu implies this condition for every orientation
(TauCeti.exists_pos_two_mul_le_sum_orientedQuiver_of_not_exists_dynkinType_iso).
The inverse form of
zigzagGradedCartanMatrix_map_aeval_neg_X_mul_preprojectiveHilbertSeries_eq_one_of_two_mul_le_sum:
under the same hypothesis on δ, the matrix Hilbert series of the preprojective algebra of the
oriented quiver, indexed by the vertices of G, is H(X) = C_G(-X)⁻¹ for C_G(q) the zigzag
graded Cartan matrix.
The inverse quantum Cartan comparison for an orientation without sinks. If o orients
the finite simple graph G so that every vertex is the tail of an arrow, then the zigzag graded
Cartan matrix of G evaluated at q = -X is a left inverse of the matrix Hilbert series of the
preprojective algebra of the oriented quiver: C_G(-X) H = I.
The Hilbert series of a preprojective algebra is the inverse zigzag Cartan matrix at
q = -X. If o orients the finite simple graph G so that every vertex is the tail of an
arrow, then the matrix Hilbert series of the preprojective algebra of the oriented quiver,
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.