The Frobenius–Perron character and dimension of a fusion category #
The Grothendieck ring TauCeti.SplitK0 C of a finite semisimple monoidal category has the basis
TauCeti.SplitK0.simpleBasis C of simple classes, and its structure constants in that basis are
multiplicities, hence nonnegative: it is a ℤ₊-ring. For a fusion category (rigid, with simple
unit, over an algebraically closed field k with finite-dimensional morphism spaces) it is
moreover transitive (TauCeti.sum_fusionCoeff_pos_right). So the Perron–Frobenius argument of
TauCeti/LinearAlgebra/Matrix/PerronFrobenius/Character.lean applies: there is exactly one ring
homomorphism SplitK0 C →+* ℝ taking nonnegative values on the simple classes. This is the
Frobenius–Perron character TauCeti.SplitK0.frobeniusPerronChar k C. Its values on the
simple classes are positive, and its value on a simple class S is the spectral radius of the
fusion matrix of S. Its value at the class of an object is the Frobenius–Perron dimension of
that object.
Every complex character is bounded in norm on simple classes by the Frobenius–Perron character, since its values are eigenvalues of the fusion matrices. The triangle inequality extends this comparison to the class of every object.
By uniqueness, the character does not depend on k: any two fields for which C satisfies the
hypotheses give the same ring homomorphism (TauCeti.SplitK0.eq_frobeniusPerronChar).
The Frobenius–Perron dimension TauCeti.frobeniusPerronDim k X of an object X is the value
of the character at the class of X. It is multiplicative on tensor products, additive on
biproducts, 1 on the tensor unit, and invariant under taking duals, since the character is
real and characters of the Grothendieck ring intertwine duality with complex conjugation
(TauCeti.SplitK0.ringHom_apply_dual). It is the sum of the dimensions of the simple summands of
X, counted with multiplicity, so it is nonnegative and monotone along monomorphisms and
epimorphisms. A nonzero object X has Frobenius–Perron dimension at least 1, because the
coevaluation embeds the unit into X ⊗ Xᘁ, whose dimension is the square of that of X. So the
Frobenius–Perron dimension vanishes exactly on zero objects.
The Frobenius–Perron dimension of the category, TauCeti.frobeniusPerronGlobalDim k C, is the
sum of the squared Frobenius–Perron dimensions of the simple classes. It is the Frobenius–Perron
counterpart of the global dimension of a pivotal structure, which it bounds in absolute value over
ℂ; the two agree exactly for pseudo-unitary categories.
Main declarations #
TauCeti.SplitK0.frobeniusPerronChar k C: the Frobenius–Perron character.TauCeti.SplitK0.frobeniusPerronChar_simpleBasis_pos: it is positive on the simple classes.TauCeti.SplitK0.eq_frobeniusPerronChar: it is the only ring homomorphism toℝwhich is nonnegative on the simple classes.TauCeti.SplitK0.spectralRadius_fusionMatrix: its value on a simple class is the spectral radius of the fusion matrix.TauCeti.frobeniusPerronDim k X: the Frobenius–Perron dimension of an object.TauCeti.frobeniusPerronDim_tensorObj,TauCeti.frobeniusPerronDim_tensorUnitandTauCeti.frobeniusPerronDim_biprod: it is multiplicative, normalized and additive.TauCeti.frobeniusPerronDim_rightDualandTauCeti.frobeniusPerronDim_leftDual: it is invariant under duality.TauCeti.frobeniusPerronDim_eq_sum: it is the sum over the simple summands.TauCeti.SplitK0.norm_ringHom_of_le_frobeniusPerronDim: the value of any complex character on an object is bounded in norm by its Frobenius–Perron dimension.TauCeti.frobeniusPerronDim_le_of_monoandTauCeti.frobeniusPerronDim_le_of_epi: it is monotone along monomorphisms and epimorphisms.TauCeti.one_le_frobeniusPerronDim,TauCeti.frobeniusPerronDim_eq_zero_iffandTauCeti.frobeniusPerronDim_pos_iff: it is at least one on nonzero objects, and vanishes exactly on zero objects.TauCeti.frobeniusPerronGlobalDim k C: the Frobenius–Perron dimension of the category, the sum of the squared Frobenius–Perron dimensions of the simple classes.
References #
- P. Etingof, S. Gelaki, D. Nikshych, V. Ostrik, Tensor Categories, Mathematical Surveys and Monographs 205, AMS (2015), Sections 3.3 (Frobenius–Perron dimensions of transitive ℤ₊-rings) and 4.5 (Frobenius–Perron dimensions in a fusion category).
Transitivity of the Grothendieck ring of a fusion category, in coordinates: every simple class occurs in the product of any basis element with the sum of all simple classes.
The Grothendieck ring of a fusion category has exactly one ring homomorphism to ℝ which is
nonnegative on the simple classes.
The Frobenius–Perron character of a fusion category: the unique ring homomorphism from
the Grothendieck ring to ℝ which is nonnegative on the simple classes
(TauCeti.SplitK0.eq_frobeniusPerronChar). Its value at the class of an object is the
Frobenius–Perron dimension of that object.
Instances For
A ring homomorphism from the Grothendieck ring of a fusion category to ℝ which is
nonnegative on the simple classes is the Frobenius–Perron character.
The Frobenius–Perron character is positive on the simple classes.
The value of the Frobenius–Perron character on a simple class S is the spectral radius of
the fusion matrix of S.
Every complex character of the Grothendieck ring is bounded on a simple class by the Frobenius–Perron character.
The Frobenius–Perron dimension of an object of a fusion category: the value of the
Frobenius–Perron character TauCeti.SplitK0.frobeniusPerronChar at the class of the object.
Instances For
The Frobenius–Perron character at the class of an object is its Frobenius–Perron dimension.
Isomorphic objects have the same Frobenius–Perron dimension.
The Frobenius–Perron dimension is multiplicative.
The tensor unit has Frobenius–Perron dimension one.
The Frobenius–Perron dimension is additive on biproducts.
A zero object has Frobenius–Perron dimension zero.
The zero object has Frobenius–Perron dimension zero.
A right dual has the same Frobenius–Perron dimension as the object.
A left dual has the same Frobenius–Perron dimension as the object.
A simple object has positive Frobenius–Perron dimension.
The Frobenius–Perron dimension of an object is the sum of the Frobenius–Perron dimensions of its simple summands, counted with multiplicity.
Every complex character of the Grothendieck ring, evaluated on an object, has norm at most the Frobenius–Perron dimension of that object.
The Frobenius–Perron dimension is nonnegative.
The Frobenius–Perron dimension is monotone along monomorphisms.
The Frobenius–Perron dimension is monotone along epimorphisms.
A nonzero object has Frobenius–Perron dimension at least one.
An object has Frobenius–Perron dimension zero exactly when it is a zero object.
An object has positive Frobenius–Perron dimension exactly when it is nonzero.
The Frobenius–Perron dimension of a fusion category: the sum, over the simple classes, of the squared Frobenius–Perron dimensions of their chosen representatives.
Instances For
The defining formula of the Frobenius–Perron dimension of a fusion category.
The Frobenius–Perron dimension of a fusion category is at least one: the class of the tensor
unit contributes 1.