Cartan coordinates of componentwise vertex projectives

The idempotent graded dimension of Z e_j has diagonal coefficients in degrees zero and two, and edge coefficients in degree one. These are the coordinates of the graded Cartan map for the public algebra, including its isolated-vertex dual-number factors.

The computation uses the generic graded-ideal/corner comparison and the actual signed corner dimensions. See Huerfano--Khovanov, A category for the adjoint representation, Section 3, equation (15), for the corner gradings and Section 6.1, equation (36), for the Cartan-polynomial convention. The generic coordinate construction is in TauCeti.Algebra.Category.GradedModuleCat.CartanMap.Idempotent.Coordinate.

theorem TauCeti.coeff_smulGradedDimension_zigzagAlgebraGradedProjective (k : Type w) [Field k] {V : Type u} [Finite V] (G : SimpleGraph V) (i j : V) (d : ℤ) : (GradedModuleCat.smulGradedDimension ((fun (i : V) => (zigzagAlgebraBasis k G) (Sum.inl i)) i) (zigzagAlgebraGradedProjective k G j)).coeff d = if i = j then if d = 0 ∨ d = 2 then 1 else 0 else if G.Adj i j ∧ d = 1 then 1 else 0

The internal-degree coefficients of a vertex projective's idempotent graded dimension.

theorem TauCeti.smulGradedDimension_zigzagAlgebraGradedProjective (k : Type w) [Field k] {V : Type u} [Finite V] (G : SimpleGraph V) (i j : V) : GradedModuleCat.smulGradedDimension ((fun (i : V) => (zigzagAlgebraBasis k G) (Sum.inl i)) i) (zigzagAlgebraGradedProjective k G j) = if i = j then 1 + LaurentPolynomial.T 2 else if G.Adj i j then LaurentPolynomial.T 1 else 0

The vertex-projective Cartan coordinate is 1 + q² on the diagonal and q on an edge.

theorem TauCeti.gradedIdempotentCoordinate_gradedCartanMap_zigzagAlgebraGradedProjectiveClass (k : Type w) [Field k] {V : Type u} [Finite V] (G : SimpleGraph V) (i j : V) : (gradedIdempotentCoordinate ⋯ ⋯) ((gradedCartanMap (zigzagAlgebraIntegerGrade k G)) (zigzagAlgebraGradedProjectiveClass k G j)) = if i = j then 1 + LaurentPolynomial.T 2 else if G.Adj i j then LaurentPolynomial.T 1 else 0

The graded Cartan map has diagonal coordinate 1 + q², edge coordinate q, and zero otherwise, for the public algebra even at isolated vertices.