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Unitary Fusion Spaces, F/R Symbols, and Anyon Models — Examples

1 · Prerequisites

2 · Summary

These worked carriers give exact Fibonacci and Ising three-anyon braid matrices, verify the B3 relation, and compute the projective generator orders. A separate one-dimensional Hecke module shows that the algebraic parameter alone does not provide a positive unitary structure.

3 · Logical flowchart

4 · Definitions, theorems and proofs

None yet.

5 · Examples, counterexamples and false statements

CounterexampleConstruction: AI-generatedVerification: AI-generatedOpen item page →

A Hecke parameter does not certify a positive unitary structure

Statement refuted

The claim refuted is that the algebraic type-A Hecke relations at every parameter automatically supply a positive-definite inner product making the resulting module action unitary.

Facts & Assumptions

Given: the root type-A Hecke algebra in its $v$ -normalization, specialized to $v = 2$ , and the one-dimensional complex vector space $V = C$ .

[F1]

The generic type-A Hecke algebra: at a unit specialization $v_{0}$ , the type-A generators satisfy $(T_{i} - v_{0}) (T_{i} + 1) = 0$ and the Artin braid and distant-commutativity relations.

[F2]

Real and complex inner-product spaces and their induced length: a positive-definite complex inner product has induced norm $∥ z ∥ = \sqrt{⟨ z, z ⟩}$ , with $∥ z ∥ > 0$ for $z \neq 0$ and $∥ λ z ∥ = ∣ λ ∣ ∥ z ∥$ .

[F3]

Linear isometries, and orthogonal or unitary operators on finite-dimensional inner product spaces: a unitary operator is an invertible linear isometry and therefore preserves this norm.

Counterexample

1.1F1givenconstructalgebra

The specialization $v \mapsto 2$ is valid because $2 \in C^{\times}$ . Assign $T_{1} \mapsto 2 I_{V}$ and $T_{2} \mapsto 2 I_{V}$ . The quadratic relator maps to $(2 - 2) (2 + 1) I_{V} = 0$ . The adjacent braid relator maps to $2 I_{V} 2 I_{V} 2 I_{V} = 8 I_{V}$ on both sides; there is no distant pair for $n = 3$ . Thus every defining relator vanishes, so the assignment from the free algebra descends to a one-dimensional $H_{2} (3)$ -module.

2.1F2F3step 1.1algebra

Let $z \neq 0$ . For any positive-definite Hermitian form on $V$ , $∥ T_{i} z ∥ = ∥ 2 z ∥ = 2 ∥ z ∥ \neq ∥ z ∥$ by [F2]. Therefore $T_{i} = 2 I_{V}$ is not an isometry and cannot be unitary by [F3], for any choice of positive inner product.

3.1F1step 1.1step 2.1∎

The specialization satisfies the algebraic Hecke and braid relations but its one-dimensional module has no positive inner product making its generators unitary. Hence an algebraic Hecke parameter alone is not a positivity or unitarity certificate. This witness says nothing about other parameters or about modules supplied with a separately verified dagger structure.

ExampleConstruction: Literature-sourcedVerification: AI-adaptedOpen item page →

Worked Fibonacci and Ising braid matrices

Example

For the Fibonacci three- $τ$ , total- $τ$ sector and the Ising three- $σ$ , total- $σ$ sector, use the F/R data of Fibonacci fusion data and chiral convention and Ising fusion data and chiral convention. In each sector the first generator is $B_{1} = diag (R_{1}, R_{2})$ and the second is $B_{2} = F B_{1} F$ . We verify both are unitary, satisfy the $B_{3}$ Artin relation, and have projective orders $10$ (Fibonacci) and $4$ (Ising).

Facts & Assumptions

Given: the two fixed two-dimensional fusion sectors and their orthonormal bases, F matrices, and R eigenvalues.

[F1]

Fibonacci fusion data and chiral convention: in the Fibonacci sector, $F = (\begin{array}{cc} u & v \\ v & - u \end{array})$ , $u = ϕ^{- 1}$ , $v = ϕ^{- 1 / 2}$ , $R_{1} = e^{- 4 π i / 5}$ , and $R_{2} = e^{3 π i / 5}$ .

[F2]

Ising fusion data and chiral convention: in the Ising sector, $F = 2^{- 1 / 2} (\begin{array}{cc} 1 & 1 \\ 1 & - 1 \end{array})$ , $R_{1} = e^{- i π / 8}$ , and $R_{2} = e^{3 i π / 8}$ .

[F3]

The braid group by Artin presentation: for three strands the defining adjacent-generator relation is $B_{1} B_{2} B_{1} = B_{2} B_{1} B_{2}$ .

[F4]

Linear isometries, and orthogonal or unitary operators on finite-dimensional inner product spaces: a unitary operator preserves the inner-product norm; in an orthonormal basis its matrix satisfies $B^{†} B = B B^{†} = I$ .

Verification

Given: the data in [F1] and [F2].

1.1F1F2F4algebra

In either sector write $B_{1} = diag (x, y)$ and $B_{2} = F B_{1} F$ . The F matrices in [F1] and [F2] are unitary by their displayed entries and the identities proved for the respective examples. Since $x, y$ are unit complex numbers, $B_{1}$ is unitary; $B_{2}$ is a product of unitary matrices and is therefore unitary as well.

1.2F1F2algebra

A projective class of $diag (x, y)$ has the same order as $r = y / x$ . In the Fibonacci case $r = e^{2 π i (- 3 / 10)}$ has exact order $10$ because $\gcd (3, 10) = 1$ . In the Ising case $r = i$ has exact order $4$ . These orders ignore only the common scalar $x$ .

2.1F1F3step 1.1algebra

In the Fibonacci case put $r = y / x = e^{- 3 π i / 5}$ and $D = diag (1, r)$ . Then $B_{1} = x D$ and $B_{2} = x C$ , where $C = F D F$ . Put $s = r^{2} + u r + 1$ . Direct multiplication gives $D C D - C D C = (\begin{array}{cc} u^{3} (1 - r) s & - u^{3} v (1 - r) s \\ - u^{3} v (1 - r) s & u^{3} (r - 1) s \end{array}) .$ Here $u^{2} + u = 1$ , $v^{2} = u$ , and $r + r^{- 1} = 2 \cos (3 π / 5) = - u$ , so $s = 0$ . Thus $D C D = C D C$ and $B_{1} B_{2} B_{1} = x^{3} D C D = x^{3} C D C = B_{2} B_{1} B_{2}$ .

2.2F2F3step 1.1algebra

In the Ising case put $r = y / x = i$ and $D = diag (1, r)$ . Then $B_{1} = x D$ and $C = F D F = \frac{1}{2} (\begin{array}{cc} 1 + r & 1 - r \\ 1 - r & 1 + r \end{array}), B_{2} = x C .$ Multiplication yields $D C D - C D C = \frac{1 + r^{2}}{4} (\begin{array}{cc} 1 - r & - (1 - r) \\ - (1 - r) & - (1 - r) \end{array}) = 0,$ because $r^{2} = - 1$ . Therefore $B_{1} B_{2} B_{1} = x^{3} D C D = x^{3} C D C = B_{2} B_{1} B_{2}$ in the Ising sector as well.

3.1F1F2F3F4step 1.1step 2.1step 2.2step 1.2∎

Steps 1.1–1.2 give the explicit unitary braid matrices, verify their Artin relation, and compute their projective generator orders in both stated sectors. No statement about other Ising sectors is used.

Sources

Chetan Nayak et al., Non-Abelian Anyons and Topological Quantum Computation