Characters of the Grothendieck ring of a fusion category are ⋆-characters

Let C be a rigid monoidal category which is k-linear abelian and finite semisimple, over an algebraically closed field k, with finite-dimensional morphism spaces and finitely many simple classes. Its Grothendieck ring TauCeti.SplitK0 C carries the duality [X] ↦ [Xᘁ]. This file shows that every ring homomorphism ψ from the Grothendieck ring to ℝ or ℂ intertwines the duality with complex conjugation: ψ [Xᘁ] = conj (ψ [X]).

The proof is the common-eigenvector argument of Etingof–Gelaki–Nikshych–Ostrik. The vector v = (ψ [T])_T of values of ψ on the simple classes is nonzero, since ψ 1 = 1, and it is an eigenvector of the fusion matrix N_S of every simple class S, with eigenvalue ψ [S] (TauCeti.SplitK0.fusionMatrix_map_mulVec_ringHom). Frobenius reciprocity says that the fusion matrix of the dual class is the transpose, N_{Sᘁ} = N_Sᵀ (TauCeti.fusionMatrix_dual), and since N_S has real entries this is its conjugate transpose. Pairing with v in the standard Hermitian form gives ψ [S] ‖v‖² = ⟨N_S v, v⟩ = ⟨v, N_{Sᘁ} v⟩ = conj (ψ [Sᘁ]) ‖v‖².

No pivotal structure and no simplicity of the tensor unit is needed: the duality is formed with right duals, and Frobenius reciprocity holds for them in any rigid category.

For instance, a real-valued character takes the same value on [X] and on [Xᘁ]; and for a complex-valued character ψ (x * dual x) = ψ x * conj (ψ x) is the squared absolute value of ψ x, the input to the nonvanishing of the global dimension of a fusion category in characteristic zero (Etingof–Nikshych–Ostrik).

Main declarations

• TauCeti.SplitK0.ringHom_apply_dual: a ring homomorphism ψ from the Grothendieck ring to ℝ or ℂ satisfies ψ (dual x) = conj (ψ x).
• TauCeti.SplitK0.ringHom_apply_simpleBasis_dual: the same identity on the basis of simple classes.

References

• [P. Etingof, S. Gelaki, D. Nikshych, V. Ostrik, Tensor categories][egno15], §§3.3, 4.5
• P. Etingof, D. Nikshych, V. Ostrik, On fusion categories, Ann. of Math. 162 (2005), proof of Theorem 2.3

theorem TauCeti.SplitK0.ringHom_apply_simpleBasis_dual (k : Type w') [Field k] [IsAlgClosed k] {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Abelian C] [CategoryTheory.MonoidalCategory C] [CategoryTheory.MonoidalPreadditive C] [CategoryTheory.RigidCategory C] [CategoryTheory.Linear k C] [CategoryTheory.MonoidalLinear k C] [IsFiniteSemisimpleCategory C] [∀ (X Y : C), FiniteDimensional k (X ⟶ Y)] [CategoryTheory.EssentiallySmall.{w, v, u} C] [Finite (SimpleClasses C)] {𝕜 : Type u_1} [RCLike 𝕜] (ψ : SplitK0 C →+* 𝕜) (S : SimpleClasses C) : ψ ((simpleBasis C) S.dual) = (starRingEnd 𝕜) (ψ ((simpleBasis C) S))

A ring homomorphism ψ from the Grothendieck ring of a fusion category to ℝ or ℂ takes the dual of a simple class to the complex conjugate: ψ [Sᘁ] = conj (ψ [S]).

theorem TauCeti.SplitK0.ringHom_apply_dual (k : Type w') [Field k] [IsAlgClosed k] {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Abelian C] [CategoryTheory.MonoidalCategory C] [CategoryTheory.MonoidalPreadditive C] [CategoryTheory.RigidCategory C] [CategoryTheory.Linear k C] [CategoryTheory.MonoidalLinear k C] [IsFiniteSemisimpleCategory C] [∀ (X Y : C), FiniteDimensional k (X ⟶ Y)] [CategoryTheory.EssentiallySmall.{w, v, u} C] [Finite (SimpleClasses C)] {𝕜 : Type u_1} [RCLike 𝕜] (ψ : SplitK0 C →+* 𝕜) (x : SplitK0 C) : ψ (dual x) = (starRingEnd 𝕜) (ψ x)

Characters of the Grothendieck ring are ⋆-characters. A ring homomorphism ψ from the Grothendieck ring of a fusion category to ℝ or ℂ intertwines the duality [X] ↦ [Xᘁ] with complex conjugation.