The matrix Hilbert series of a preprojective algebra #
Let Q be a finite quiver and Π = Π_k(Q) its preprojective algebra over a field k. For vertices
i and j and a degree n, the graded corner e_j Π_n e_i (TauCeti.preprojectiveCorner) is the
span of the classes of the paths of length n from i to j in the doubled quiver. Its dimensions
form the matrix Hilbert series
H(t)_{j,i} = ∑_n dim (e_j Π_n e_i) tⁿ (TauCeti.preprojectiveHilbertSeries).
Restricting the Koszul complex of the vertex module S_v (see
TauCeti.RepresentationTheory.Quiver.Preprojective.Koszul.Complex) to degree m + 2 and cutting it
down on the right by e_i gives the sequence
0 ⟶ K_m(i, v) ⟶ e_v Π_m e_i ⟶ ⨁_{b : w ⟶ v} e_w Π_{m+1} e_i ⟶ e_v Π_{m+2} e_i ⟶ 0,
where K_m(i, v) is the space of y ∈ e_v Π_m e_i killed by the reverse b* of every arrow b
of the doubled quiver into v. It is exact by the exactness of the Koszul complex at its two
right-hand terms, which holds for every quiver. Counting dimensions gives the Koszul defect
formula (TauCeti.finrank_preprojectiveCorner_add_two_add_finrank_preprojectiveCorner)
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 + dim K_m(i, v),
together with dim e_j Π_0 e_i = δ_{ij} and dim e_j Π_1 e_i = #(i ⟶ j). In matrix form, with A
the arrow-count matrix of the doubled quiver and K(t) the matrix of the series
∑_m dim K_m(i, v) tᵐ, this reads ((1 + t²) I - t A) H(t) = I + t² K(t). Consequently
(TauCeti.mul_preprojectiveHilbertSeries_eq_one_iff) the Hilbert series identity
((1 + t²) I - t A) H(t) = I
holds exactly when the left-hand map y ↦ (ε_b b* y)_b of every Koszul complex is injective, that
is, exactly when these complexes are projective resolutions of the vertex modules. This is the
form in which Etingof and Eu relate the Koszulity of Π to its Hilbert series
H(t) = ((1 + t²) I - t A)⁻¹, which they establish for connected non-Dynkin quivers; this file
proves the equivalence for every finite quiver, not the injectivity itself. When Q is an
orientation of a simple graph, (1 + t²) I - t A is the graded Cartan matrix (1 + q²) I + q A of
the zigzag algebra of that graph evaluated at q = -t.
Main definitions #
TauCeti.preprojectiveCorner: the graded cornere_j Π_n e_i.TauCeti.preprojectiveHilbertSeries: the matrix Hilbert series ofΠ.
Main results #
TauCeti.mem_preprojectiveCorner_iffandTauCeti.preprojectiveCorner_eq_cornerSubmodule_inf_preprojectiveGrade: the graded corner is the intersection of the degree-npiece with the cornere_j Π e_i.TauCeti.finrank_preprojectiveCorner_zeroandTauCeti.finrank_preprojectiveCorner_one: the dimensions in degrees0and1.TauCeti.exists_eq_sum_preprojectiveMk_ofArrow_mul_of_mem_preprojectiveCornerandTauCeti.exists_mem_preprojectiveCorner_of_sum_preprojectiveMk_ofArrow_mul_eq_zero: exactness of the Koszul complex at its two right-hand terms, degree by degree.TauCeti.eq_zero_of_forall_mem_preprojectiveCorner: exactness at the left end may be checked on the graded corners.TauCeti.finrank_preprojectiveCorner_add_two_add_finrank_preprojectiveCorner: the Koszul defect formula.TauCeti.sum_card_mul_finrank_preprojectiveCorner_eq_iff: the dimension recursion holds in degreem + 2exactly when the Koszul complex is exact at its left end in degreem.TauCeti.mul_preprojectiveHilbertSeries_eq_one_iff: the Hilbert series identity((1 + t²) I - t A) H(t) = Iholds exactly when every Koszul complex is a resolution.
References #
- P. Etingof and C.-H. Eu, Koszulity and the Hilbert series of preprojective algebras, Math. Res. Lett. 14 (2007), Sections 2 and 3.
- D. J. Anick, Non-commutative graded algebras and their Hilbert series, J. Algebra 78 (1982).
The graded corner e_j Π_n e_i of the preprojective algebra: the span of the classes of the
paths of length n from i to j in the doubled quiver. By TauCeti.mem_preprojectiveCorner_iff
it is the intersection of the degree-n piece TauCeti.preprojectiveGrade with the corner of the
two vertex idempotents.
Instances For
The graded corner e_j Π_n e_i is the degree-n part of the corner e_j Π e_i.
The graded corner as an intersection: e_j Π_n e_i is the corner submodule of the two
vertex idempotents intersected with the degree-n piece.
An element of e_j Π_n e_i is fixed by left multiplication by e_j.
An element of e_j Π_n e_i is fixed by right multiplication by e_i.
Left multiplication by an arrow b : j ⟶ l of the doubled quiver maps e_j Π_n e_i into
e_l Π_{n+1} e_i.
Distinct graded corners meet only in 0: e_j Π_m e_i and e_{j'} Π_n e_{i'} are disjoint
unless m = n, i = i' and j = j'.
The graded corners of the preprojective algebra of a finite quiver are finite-dimensional: there are finitely many paths of each length between two vertices.
Every element of e_v Π_{n+1} e_i is a sum of arrow multiples ∑_b b z_b with
z_b ∈ e_w Π_n e_i, the sum running over the arrows b : w ⟶ v of the doubled quiver. This is the
exactness of the Koszul complex at e_v Π in one degree, cut down on the right by e_i.
Exactness of the Koszul complex at its middle term, in one degree and cut down by e_i.
If z_b ∈ e_w Π_{m+1} e_i for the arrows b : w ⟶ v of the doubled quiver and ∑_b b z_b = 0,
then there is a single y ∈ e_v Π_m e_i with z_b = ε_b b* y for every b.
Exactness of the Koszul complex at its left end can be checked in each degree. If for every
degree m and every vertex i the only y ∈ e_v Π_m e_i killed by the reverse b* of every
arrow b of the doubled quiver into v is 0, then the only such y ∈ e_v Π is 0.
The graded corners in degree 0: e_j Π_0 e_i is the line spanned by e_i when i = j,
and is zero otherwise.
The graded corners in degree 1: e_j Π_1 e_i has the arrows i ⟶ j of the doubled
quiver as a basis.
The Koszul defect formula. For vertices i and v and every degree m,
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 + dim K_m(i, v),
the sum running over the arrows of the doubled quiver into v, where K_m(i, v) is the space of
y ∈ e_v Π_m e_i killed by the reverse b* of every arrow b into v, the kernel of the left
map of the Koszul complex of S_v.
The dimension recursion in degree m + 2 is Koszul exactness in degree m. The equality
∑_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 holds exactly when no
nonzero y ∈ e_v Π_m e_i is killed by the reverse b* of every arrow b of the doubled quiver
into v.
The matrix Hilbert series of the preprojective algebra: its entry in row j and column i
is ∑_n dim (e_j Π_n e_i) Xⁿ, the Hilbert series of the paths from i to j.
Instances For
The coefficient of Xⁿ in the entry of the Hilbert series in row j and column i is
dim (e_j Π_n e_i).
The Hilbert series identity characterizes injectivity of the left Koszul map. Let A be
the arrow-count matrix of the doubled quiver, A_{v,w} = #(w ⟶ v), and H the matrix Hilbert
series of Π. Then
((1 + X²) I - X A) H = I
holds 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 the left-hand map of the
Koszul complex of every vertex module is injective, so that these complexes are projective
resolutions.
The Koszul defect is nonnegative. Let A be the arrow-count matrix of the doubled quiver,
A_{v,w} = #(w ⟶ v), and H the matrix Hilbert series of Π. Every coefficient of every entry of
((1 + X²) I - X A) H is at least the corresponding coefficient of the identity matrix. Indeed the
two agree in degrees 0 and 1, and in degree m + 2 their difference is the dimension of the
space K_m(i, v) of y ∈ e_v Π_m e_i killed by the reverse b* of every arrow b into v.