The projective q-Hom form of a zigzag algebra

For a finite simple graph without isolated vertices, this file packages the homogeneous homomorphisms between zigzag vertex projectives into Laurent polynomials. With the internal shift convention M{d}_p = M_{p-d}, the value

qHom(P_i, P_j) = ∑_d dim_k Hom(P_i, P_j{d}) q⁻ᵈ

has finite support: target shifts outside [-2, 0] vanish. It is the Laurent image of the graded Cartan entry, hence its matrix is (1 + q²)I + qA_G.

The matrix also defines a sesquilinear form on the free Laurent module with basis the vertices. The Laurent involution q ↦ q⁻¹ acts in the first variable, as shifting the source of a Hom space reverses the shift. This is the coordinate form of the projective q-Hom pairing; it does not assert a presentation of the full graded Grothendieck group.

Main definitions

• TauCeti.zigzagProjectiveQHom: the Laurent-valued graded dimension of Hom(P_i, P_j{d}).
• TauCeti.zigzagProjectiveQHomMatrix: its matrix in the vertex-projective basis.
• TauCeti.zigzagProjectiveQHomForm: the resulting sesquilinear form on vertex coordinates.

Main results

• TauCeti.zigzagProjectiveQHom_eq_toLaurent: q-Hom is the Laurent image of the graded Cartan entry.
• TauCeti.zigzagProjectiveQHomMatrix_apply: its entries are 1 + q² on the diagonal, q on edges, and zero elsewhere.
• TauCeti.zigzagProjectiveQHomForm_single: the form on two vertex basis vectors is the corresponding projective q-Hom value.

References

See Huerfano--Khovanov, A category for the adjoint representation, Section 3, and Ehrig--Tubbenhauer, Algebraic properties of zigzag algebras, Section 2, for the quantum Cartan matrix of a zigzag algebra.

Finite support and the q-Hom polynomial

theorem TauCeti.hasFiniteLaurentSupport_zigzagProjectiveTargetShiftHom (k : Type w) [Field k] {V : Type u} (G : SimpleGraph V) [Finite V] (hns : ∀ (i : V), ∃ (j : V), G.Adj i j) (i j : V) : HasFiniteLaurentSupport k fun (d : ℤ) => ↥(zigzagProjectiveTargetShiftHom k G i j d)

The target-shifted homomorphism spaces between two zigzag vertex projectives have finite Laurent support. More precisely, only shifts -2, -1, and 0 can contribute.

noncomputable def TauCeti.zigzagProjectiveQHom (k : Type w) [Field k] {V : Type u} (G : SimpleGraph V) [Finite V] (hns : ∀ (i : V), ∃ (j : V), G.Adj i j) (i j : V) : LaurentPolynomial ℤ

The projective q-Hom value between the zigzag vertex projectives P_i and P_j: the Laurent polynomial ∑_d dim_k Hom(P_i, P_j{d}) q⁻ᵈ.

@[simp]

theorem TauCeti.coeff_zigzagProjectiveQHom (k : Type w) [Field k] {V : Type u} (G : SimpleGraph V) [Finite V] (hns : ∀ (i : V), ∃ (j : V), G.Adj i j) (i j : V) (n : ℤ) : (zigzagProjectiveQHom k G hns i j).coeff n = ↑(Module.finrank k ↥(zigzagProjectiveTargetShiftHom k G i j (-n)))

The coefficient of qⁿ in projective q-Hom is the dimension of the target-shifted Hom space with shift -n.

theorem TauCeti.coeff_zigzagProjectiveQHom_ofNat (k : Type w) [Field k] {V : Type u} (G : SimpleGraph V) [Finite V] (hns : ∀ (i : V), ∃ (j : V), G.Adj i j) (i j : V) (n : ℕ) : (zigzagProjectiveQHom k G hns i j).coeff ↑n = ↑(Module.finrank k ↥(zigzagProjectiveHomOfDegree k G i j n))

At a nonnegative exponent, projective q-Hom records the degree-raising Hom space Hom(P_i{n}, P_j).

theorem TauCeti.zigzagProjectiveQHom_eq_toLaurent (k : Type w) [Field k] {V : Type u} (G : SimpleGraph V) [Finite V] (hns : ∀ (i : V), ∃ (j : V), G.Adj i j) (i j : V) : zigzagProjectiveQHom k G hns i j = Polynomial.toLaurent (zigzagGradedCartanMatrix k G i j)

Projective q-Hom is the Laurent image of the graded Cartan entry.

Matrix and sesquilinear form

noncomputable def TauCeti.zigzagProjectiveQHomMatrix (k : Type w) [Field k] {V : Type u} (G : SimpleGraph V) [Finite V] (hns : ∀ (i : V), ∃ (j : V), G.Adj i j) : Matrix V V (LaurentPolynomial ℤ)

The matrix of projective q-Hom values in the vertex-projective basis.

@[simp]

theorem TauCeti.zigzagProjectiveQHomMatrix_apply_eq_qHom (k : Type w) [Field k] {V : Type u} (G : SimpleGraph V) [Finite V] (hns : ∀ (i : V), ∃ (j : V), G.Adj i j) (i j : V) : zigzagProjectiveQHomMatrix k G hns i j = zigzagProjectiveQHom k G hns i j

An entry of the projective q-Hom matrix is the q-Hom value of the corresponding pair of vertex projectives.

theorem TauCeti.zigzagProjectiveQHomMatrix_eq_map_toLaurent (k : Type w) [Field k] {V : Type u} (G : SimpleGraph V) [Finite V] (hns : ∀ (i : V), ∃ (j : V), G.Adj i j) : zigzagProjectiveQHomMatrix k G hns = (zigzagGradedCartanMatrix k G).map ⇑Polynomial.toLaurent

The q-Hom matrix is the entrywise Laurent image of the graded Cartan matrix.

theorem TauCeti.zigzagProjectiveQHomMatrix_apply (k : Type w) [Field k] {V : Type u} (G : SimpleGraph V) [Finite V] (hns : ∀ (i : V), ∃ (j : V), G.Adj i j) [DecidableEq V] [DecidableRel G.Adj] (i j : V) : zigzagProjectiveQHomMatrix k G hns i j = (if i = j then 1 + LaurentPolynomial.T 2 else 0) + if G.Adj i j then LaurentPolynomial.T 1 else 0

The entrywise projective q-Hom formula. It is 1 + q² on the diagonal, q on an edge, and zero otherwise.

theorem TauCeti.zigzagProjectiveQHomMatrix_eq (k : Type w) [Field k] {V : Type u} (G : SimpleGraph V) [Finite V] (hns : ∀ (i : V), ∃ (j : V), G.Adj i j) [DecidableEq V] [DecidableRel G.Adj] : zigzagProjectiveQHomMatrix k G hns = (1 + LaurentPolynomial.T 2) • 1 + LaurentPolynomial.T 1 • SimpleGraph.adjMatrix (LaurentPolynomial ℤ) G

The projective q-Hom matrix is (1 + q²)I + qA_G.

noncomputable def TauCeti.zigzagProjectiveQHomForm (k : Type w) [Field k] {V : Type u} (G : SimpleGraph V) [Finite V] (hns : ∀ (i : V), ∃ (j : V), G.Adj i j) [DecidableEq V] [Fintype V] : (V → LaurentPolynomial ℤ) →ₛₗ[LaurentPolynomial.invert.toRingEquiv.toRingHom] (V → LaurentPolynomial ℤ) →ₗ[LaurentPolynomial ℤ] LaurentPolynomial ℤ

The projective q-Hom form on Laurent-polynomial vertex coordinates. It is semilinear in the first variable for q ↦ q⁻¹ and linear in the second.

@[simp]

theorem TauCeti.zigzagProjectiveQHomForm_single (k : Type w) [Field k] {V : Type u} (G : SimpleGraph V) [Finite V] (hns : ∀ (i : V), ∃ (j : V), G.Adj i j) [DecidableEq V] [Fintype V] (i j : V) : ((zigzagProjectiveQHomForm k G hns) (Pi.single i 1)) (Pi.single j 1) = zigzagProjectiveQHom k G hns i j

On vertex basis vectors, the projective q-Hom form is the q-Hom value of the corresponding vertex projectives.