The Frobenius–Perron character of a transitive ℤ₊-ring

Let R be a ring with a finite nonempty ℤ-basis b whose structure constants are nonnegative: every product b i * b j is a nonnegative integer combination of basis elements (a ℤ₊-ring in the sense of Etingof–Gelaki–Nikshych–Ostrik). Suppose it is transitive on one side: every basis element occurs in b j * s for every j, where s = ∑ i, b i is the sum of the basis. Then R has exactly one character χ : R →+* ℝ taking nonnegative values on the basis, and its values on the basis are strictly positive. Its value at an element x with nonnegative coordinates is the spectral radius of the matrix of left multiplication by x.

This is the character that defines the Frobenius–Perron dimension of a fusion category, whose Grothendieck ring is a transitive ℤ₊-ring with basis the classes of the simple objects. The character cannot be defined from each multiplication matrix separately: such a matrix may be reducible (the matrix of the unit is the identity), so its nonnegative eigenvectors need not be proportional and do not single out a value for the character, and its spectral radius alone is not visibly additive or multiplicative. The character is instead constructed as below, and only afterwards is its value at an element with nonnegative coordinates identified with the spectral radius of the corresponding left-multiplication matrix.

The construction follows Etingof–Gelaki–Nikshych–Ostrik. The matrix of right multiplication by s has strictly positive entries, so by the Perron–Frobenius theorem it has an eigenvector v with positive entries whose eigenline contains every eigenvector for the same eigenvalue. Left multiplications commute with right multiplications, so each of them preserves that eigenline, and the scalar by which left multiplication by x acts on v is the character. Uniqueness comes from the transposed matrix: the values of any character on the basis form an eigenvector of it, so when they are nonnegative they are positive, and any two such eigenvectors are proportional.

Main results

• Module.Basis.exists_ringHom_apply_pos: a transitive ℤ₊-ring of finite nonzero rank has a character which is positive on the basis.
• Module.Basis.existsUnique_ringHom_nonneg: it is the only character which is nonnegative on the basis.
• Module.Basis.ringHom_eq_of_nonneg and Module.Basis.ringHom_apply_pos_of_nonneg: two characters which are nonnegative on the basis are equal, and such a character is positive on the basis. These need only transitivity, not the nonnegativity of the structure constants.
• Module.Basis.spectralRadius_leftMulMatrix: its value at an element with nonnegative coordinates is the spectral radius of the matrix of left multiplication by that element.

References

• P. Etingof, S. Gelaki, D. Nikshych, V. Ostrik, Tensor Categories, Mathematical Surveys and Monographs 205, AMS (2015), Section 3.3 (Proposition 3.3.6: Frobenius–Perron dimensions of a transitive ℤ₊-ring).

theorem Module.Basis.ringHom_apply_pos_of_nonneg {ι : Type u_1} {R : Type u_2} [Fintype ι] [Ring R] (b : Basis ι ℤ R) (ht : ∀ (j l : ι), 0 < ∑ i : ι, (b.repr (b j * b i)) l) {χ : R →+* ℝ} (hχ : ∀ (i : ι), 0 ≤ χ (b i)) (i : ι) : 0 < χ (b i)

If every basis element occurs in b j * ∑ i, b i for every j, then a character to ℝ which is nonnegative on the basis is positive on the basis.

theorem Module.Basis.ringHom_eq_of_nonneg {ι : Type u_1} {R : Type u_2} [Fintype ι] [Ring R] (b : Basis ι ℤ R) (ht : ∀ (j l : ι), 0 < ∑ i : ι, (b.repr (b j * b i)) l) {χ ψ : R →+* ℝ} (hχ : ∀ (i : ι), 0 ≤ χ (b i)) (hψ : ∀ (i : ι), 0 ≤ ψ (b i)) : χ = ψ

Uniqueness of the Frobenius–Perron character. If every basis element occurs in b j * ∑ i, b i for every j, then two characters to ℝ which are nonnegative on the basis are equal.

theorem Module.Basis.exists_ringHom_apply_pos {ι : Type u_1} {R : Type u_2} [Fintype ι] [Ring R] (b : Basis ι ℤ R) (hb : ∀ (i j l : ι), 0 ≤ (b.repr (b i * b j)) l) (ht : ∀ (j l : ι), 0 < ∑ i : ι, (b.repr (b j * b i)) l) [Nonempty ι] : ∃ (χ : R →+* ℝ), ∀ (i : ι), 0 < χ (b i)

A ring with a finite nonempty ℤ-basis with nonnegative structure constants, such that every basis element occurs in b j * ∑ i, b i for every j, has a character to ℝ taking positive values on the basis.

theorem Module.Basis.existsUnique_ringHom_nonneg {ι : Type u_1} {R : Type u_2} [Fintype ι] [Ring R] (b : Basis ι ℤ R) (hb : ∀ (i j l : ι), 0 ≤ (b.repr (b i * b j)) l) (ht : ∀ (j l : ι), 0 < ∑ i : ι, (b.repr (b j * b i)) l) [Nonempty ι] : ∃! χ : R →+* ℝ, ∀ (i : ι), 0 ≤ χ (b i)

The Frobenius–Perron character. A ring with a finite nonempty ℤ-basis with nonnegative structure constants, such that every basis element occurs in b j * ∑ i, b i for every j, has exactly one character to ℝ which is nonnegative on the basis.

theorem Module.Basis.spectralRadius_leftMulMatrix {ι : Type u_1} {R : Type u_2} [Fintype ι] [Ring R] (b : Basis ι ℤ R) (hb : ∀ (i j l : ι), 0 ≤ (b.repr (b i * b j)) l) (ht : ∀ (j l : ι), 0 < ∑ i : ι, (b.repr (b j * b i)) l) [DecidableEq ι] {χ : R →+* ℝ} (hχ : ∀ (i : ι), 0 ≤ χ (b i)) {x : R} (hx : ∀ (i : ι), 0 ≤ (b.repr x) i) : spectralRadius ℂ (((Algebra.leftMulMatrix b) x).map Int.cast) = ENNReal.ofReal (χ x)

The Frobenius–Perron character is a spectral radius. If χ is a character of a transitive ℤ₊-ring which is nonnegative on the basis, then for every x with nonnegative coordinates, χ x is the spectral radius of the matrix of left multiplication by x.