Documentation

TauCeti.RepresentationTheory.Quiver.Zigzag.Braid.K0

The zigzag braid complexes on the graded Grothendieck group #

The braid complex B_i = [U_i ⟶ Z] of the zigzag algebra Z, with U_i = P_i ⊗_k e_i Z in cohomological degree -1, sends the graded vertex projective P_j to a two-term complex of finite graded projectives. Its Euler class in the Laurent Grothendieck group K₀^gr(proj Z) is

[Z ⊗_Z P_j] - [U_i ⊗_Z P_j] = [P_j] - C_{ij}(q) [P_i],

where C_{ij}(q) is the graded Cartan entry: 1 + q² on the diagonal, q on edges and 0 otherwise. This is the deformed simple reflection SimpleGraph.deformedReflection at the unit q, written in the basis of vertex-projective classes. Hence the Artin group of the graph acts on K₀^gr(proj Z) with the standard generator σ_i acting by the Euler class of B_i, and this action is conjugate, through the vertex-projective coordinates, to the deformed reflection representation SimpleGraph.deformedReflectionRepresentation. At q = -1 the deformed reflections are the standard simple reflections of the graph Cartan matrix 2I - A (SimpleGraph.deformedReflection_neg_one).

Main definitions #

Main results #

References #

The terms of B_i ⊗_Z P_j are finite graded projectives #

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theorem TauCeti.gradedFiniteProjectiveModules_zigzagBraidProjectiveTensor (k : Type w) {V : Type u} [Finite V] (G : SimpleGraph V) [CommRing k] (i j : V) :
gradedFiniteProjectiveModules (zigzagAlgebraIntegerGrade k G) (GradedModuleCat.bimoduleModuleTensorObj (zigzagAlgebraInternalGrading k G) (zigzagAlgebraInternalGrading k G) (zigzagGradedBraidBimodule k G i) (zigzagAlgebraGradedProjective k G j))

The degree -1 term U_i ⊗_Z P_j of B_i ⊗_Z P_j is a finite graded projective.

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theorem TauCeti.gradedFiniteProjectiveModules_zigzagBraidProjectiveUnitTensor (k : Type w) {V : Type u} [Finite V] (G : SimpleGraph V) [CommRing k] (j : V) :
gradedFiniteProjectiveModules (zigzagAlgebraIntegerGrade k G) (GradedModuleCat.bimoduleModuleTensorObj (zigzagAlgebraInternalGrading k G) (zigzagAlgebraInternalGrading k G) (zigzagGradedBimodule k G) (zigzagAlgebraGradedProjective k G j))

The degree 0 term Z ⊗_Z P_j of B_i ⊗_Z P_j is a finite graded projective.

Their classes #

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theorem TauCeti.laurentK0_of_zigzagBraidProjectiveUnitTensor (k : Type w) {V : Type u} [Finite V] (G : SimpleGraph V) [CommRing k] (j : V) :
LaurentK0.of (gradedFiniteProjectiveModulesExactStructure (zigzagAlgebraIntegerGrade k G)) { obj := GradedModuleCat.bimoduleModuleTensorObj (zigzagAlgebraInternalGrading k G) (zigzagAlgebraInternalGrading k G) (zigzagGradedBimodule k G) (zigzagAlgebraGradedProjective k G j), property := ⋯ } = zigzagAlgebraGradedProjectiveClass k G j

[Z ⊗_Z P_j] = [P_j] in K₀^gr(proj Z).

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theorem TauCeti.laurentK0_of_zigzagBraidProjectiveTensor (k : Type w) {V : Type u} [Finite V] (G : SimpleGraph V) [CommRing k] [DecidableEq V] (i j : V) :
LaurentK0.of (gradedFiniteProjectiveModulesExactStructure (zigzagAlgebraIntegerGrade k G)) { obj := GradedModuleCat.bimoduleModuleTensorObj (zigzagAlgebraInternalGrading k G) (zigzagAlgebraInternalGrading k G) (zigzagGradedBraidBimodule k G i) (zigzagAlgebraGradedProjective k G j), property := ⋯ } = G.deformedCartanEntry (LaurentPolynomial.T 1) i j • zigzagAlgebraGradedProjectiveClass k G i

[U_i ⊗_Z P_j] = C_{ij}(q) [P_i] in K₀^gr(proj Z), where C_{ij}(q) is the deformed Cartan entry 1 + q², q or 0 at q = T 1.

The Artin group action #

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noncomputable def TauCeti.zigzagBraidK0Action (k : Type w) {V : Type u} [Finite V] (G : SimpleGraph V) [Field k] [Fintype V] [DecidableEq V] :
ArtinGroup G.coxeterMatrix →* LaurentK0 (gradedFiniteProjectiveModulesExactStructure (zigzagAlgebraIntegerGrade k G)) ≃ₗ[LaurentPolynomial ℤ] LaurentK0 (gradedFiniteProjectiveModulesExactStructure (zigzagAlgebraIntegerGrade k G))

The action of the graph's Artin group on K₀^gr(proj Z). The standard generator at i acts by the deformed simple reflection at q, in the basis of vertex-projective classes; by TauCeti.zigzagBraidK0Action_gen_zigzagAlgebraGradedProjectiveClass it sends [P_j] to the Euler class of the braid complex B_i ⊗_Z P_j.

Instances For
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    theorem TauCeti.zigzagBraidK0Action_gen_apply (k : Type w) {V : Type u} [Finite V] (G : SimpleGraph V) [Field k] [Fintype V] [DecidableEq V] (i : V) (x : LaurentK0 (gradedFiniteProjectiveModulesExactStructure (zigzagAlgebraIntegerGrade k G))) :
    ((zigzagBraidK0Action k G) (ArtinGroup.gen G.coxeterMatrix i)) x = (zigzagAlgebraGradedProjectiveClassBasis k G).equivFun.symm ((G.deformedReflection ⋯.unit i) ((zigzagAlgebraGradedProjectiveClassBasis k G).equivFun x))

    The standard generator at i acts by the deformed reflection at q in vertex-projective coordinates.

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    theorem TauCeti.zigzagBraidK0Action_gen_zigzagAlgebraGradedProjectiveClass (k : Type w) {V : Type u} [Finite V] (G : SimpleGraph V) [Field k] [Fintype V] [DecidableEq V] (i j : V) :
    ((zigzagBraidK0Action k G) (ArtinGroup.gen G.coxeterMatrix i)) (zigzagAlgebraGradedProjectiveClass k G j) = LaurentK0.of (gradedFiniteProjectiveModulesExactStructure (zigzagAlgebraIntegerGrade k G)) { obj := GradedModuleCat.bimoduleModuleTensorObj (zigzagAlgebraInternalGrading k G) (zigzagAlgebraInternalGrading k G) (zigzagGradedBimodule k G) (zigzagAlgebraGradedProjective k G j), property := ⋯ } - LaurentK0.of (gradedFiniteProjectiveModulesExactStructure (zigzagAlgebraIntegerGrade k G)) { obj := GradedModuleCat.bimoduleModuleTensorObj (zigzagAlgebraInternalGrading k G) (zigzagAlgebraInternalGrading k G) (zigzagGradedBraidBimodule k G i) (zigzagAlgebraGradedProjective k G j), property := ⋯ }

    The braid complex acts on K₀^gr(proj Z) by its Euler class: the standard generator at i sends [P_j] to [Z ⊗_Z P_j] - [U_i ⊗_Z P_j], the Euler class of B_i ⊗_Z P_j, whose terms sit in cohomological degrees 0 and -1.