Frobenius–Schur indicators of pivotal categories #
In a pivotal category C, a morphism f : 𝟙_ C ⟶ X ⊗ Y can be rotated to a morphism
pivotalRotate f : 𝟙_ C ⟶ Y ⊗ X by bending the strand X around to the right: close it up
against Xᘁ with the evaluation ε_ X Xᘁ, and reopen it on the right with the coevaluation of
the pivotal pairing TauCeti.pivotalExactPairing X : ExactPairing Xᘁ X. This is the map E_{X,Y}
of Ng and Schauenburg; with the pivotal isomorphism φ_X : X ≅ Xᘁᘁ written out, it is
(ε_ X Xᘁ ⊗ 𝟙 ⊗ φ_X⁻¹) ∘ (𝟙 ⊗ f ⊗ 𝟙) ∘ η_ Xᘁ Xᘁᘁ, up to associators and unitors.
The rotation is natural in Y and dinatural in X. Rotating a block X ⊗ X' is rotating X and
then X' (TauCeti.pivotalRotate_tensor), which is where the monoidality of the pivotal structure
enters, and rotating the unit object, on either side, changes nothing but unitors.
For an object V and n : ℕ, rotating the first factor of V^⊗(n+1) = V ⊗ V^⊗n to the end and
re-bracketing is the endomorphism TauCeti.frobeniusSchurRotate V n of 𝟙_ C ⟶ V^⊗(n+1), the
operator E_V^{(n+1)} of Ng and Schauenburg. Its (n + 1)-st iterate is the identity
(TauCeti.iterate_frobeniusSchurRotate_eq_id): by the block rule, rotating the factors one at a
time n + 1 times is rotating the whole of V^⊗(n+1) past the unit, which is trivial.
In a k-linear pivotal category the operator is k-linear, and the n-th Frobenius–Schur
indicator of V, for a positive integer n, is its trace
TauCeti.frobeniusSchurIndicator k V n = tr (E_V^{(n)}) ∈ k. This is the Ng–Schauenburg indicator
when 𝟙_ C ⟶ V^⊗n is finite and free over k (for example over a field, with finite-dimensional
hom spaces); otherwise it is 0, by the convention of LinearMap.trace. It is invariant under
isomorphism, and ν_1(V) is the rank of 𝟙_ C ⟶ V when that module is finite and free.
Main declarations #
TauCeti.pivotalRotate: the rotation(𝟙_ C ⟶ X ⊗ Y) → (𝟙_ C ⟶ Y ⊗ X), withTauCeti.pivotalRotate_comp_whiskerLeft,TauCeti.pivotalRotate_comp_whiskerRight,TauCeti.pivotalRotate_tensor,TauCeti.pivotalRotate_unit_leftandTauCeti.pivotalRotate_unit_right;TauCeti.frobeniusSchurRotate: the operatorE_V^{(n+1)}on𝟙_ C ⟶ tensorPow V (n + 1), withTauCeti.iterate_frobeniusSchurRotate_eq_idand its naturalityTauCeti.frobeniusSchurRotate_comp_tensorPowMap;TauCeti.frobeniusSchurEnd: the same operator as ak-linear endomorphism, of order dividingn + 1(TauCeti.frobeniusSchurEnd_pow_eq_one);TauCeti.frobeniusSchurIndicator: the indicatorν_n(V), withTauCeti.frobeniusSchurIndicator_congr,TauCeti.frobeniusSchurIndicator_one, andTauCeti.frobeniusSchurIndicator_two.
References #
- S.-H. Ng, P. Schauenburg, Higher Frobenius–Schur indicators for pivotal categories, in
Hopf algebras and generalizations, Contemp. Math. 441 (2007), 63–90, §§4–5, for the
rotation
E_{X,Y}, the operatorE_V^{(n)}, the identity(E_V^{(n)})^n = idand the indicators - S.-H. Ng, P. Schauenburg, Frobenius–Schur indicators and exponents of spherical categories, Adv. Math. 211 (2007), 34–71, §5
The rotation of a morphism f : 𝟙_ C ⟶ X ⊗ Y in a pivotal category to a morphism
𝟙_ C ⟶ Y ⊗ X: the strand X is closed up against Xᘁ by the evaluation ε_ X Xᘁ and reopened
on the right by the coevaluation of the pivotal pairing pivotalExactPairing X.
Instances For
The defining formula of the rotation.
Rotating the coevaluation η_ X Xᘁ : 𝟙_ C ⟶ X ⊗ Xᘁ yields the coevaluation of
pivotalExactPairing X.
The rotation is natural in the second factor.
The rotation is dinatural in the first factor: for a : X ⟶ X', rotating f ≫ a ▷ Y past
X' is rotating f past X and then applying a.
Rotating a tensor product: rotating the block X ⊗ X' to the end is rotating X and then
X', up to associators.
Rotating the unit object to the end changes only unitors.
Rotating an object past the unit object changes only unitors.
The operator E_V^{(n+1)} of Ng and Schauenburg on 𝟙_ C ⟶ V^⊗(n+1): rotate the first factor
of V^⊗(n+1) = V ⊗ V^⊗n to the end, then re-bracket V^⊗n ⊗ V as V^⊗(n+1).
Instances For
The defining formula of E_V^{(n+1)}.
Ng–Schauenburg: the (n + 1)-st iterate of E_V^{(n+1)} is the identity.
E_V^{(n+1)} is natural in V: it commutes with the tensor powers of a morphism.
The rotation is additive.
The rotation is k-linear.
The operator E_V^{(n+1)} as a k-linear endomorphism of 𝟙_ C ⟶ V^⊗(n+1).
Instances For
The linear operator frobeniusSchurEnd k V n is frobeniusSchurRotate V n.
The (n + 1)-st power of E_V^{(n+1)} is the identity.
The n-th Frobenius–Schur indicator ν_n(V) of an object V of a k-linear pivotal
category, for a positive integer n: the trace of the operator E_V^{(n)} on 𝟙_ C ⟶ V^⊗n.
This is the Ng–Schauenburg indicator when 𝟙_ C ⟶ V^⊗n is finite and free over k (for example
when k is a field and the hom spaces are finite-dimensional). Otherwise it is 0, by the
convention of LinearMap.trace.
Instances For
The indicator ν_{n+1}(V) is the trace of frobeniusSchurEnd k V n.
The second Frobenius–Schur indicator ν₂(V) is the trace of frobeniusSchurEnd k V 1.
The first Frobenius–Schur indicator is the dimension of 𝟙_ C ⟶ V, since E_V^{(1)} is the
identity.
The Frobenius–Schur indicators of isomorphic objects agree.