TheoremStatement: Literature-sourcedProof: AI-generatedPipeline-generatedprecheck passverified 2026-10-22 (gpt-6.1-sol)

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Temperley–Lieb projectors give a Hecke braid representation

Statement

For a complex vector space $V$ , let ${End}_{C} (V)$ be its set of complex-linear endomorphisms, with composition as multiplication, and define $GL (V)$ to be the group of bijective elements of ${End}_{C} (V)$ . A complex Hilbert space is a complete complex inner-product space; here the inner product is linear in its first variable and $∥ x ∥ : = \sqrt{⟨ x, x ⟩}$ . A linear operator $A : V \to V$ on such a space is bounded if some $C < \infty$ satisfies $∥ A x ∥ \leq C ∥ x ∥$ for every $x \in V$ , self-adjoint if $⟨ A x, y ⟩ = ⟨ x, A y ⟩$ for all $x, y \in V$ , and unitary if it is surjective and preserves the inner product. In the formulas below, $I_{V}$ denotes the identity map of $V$ .

Let $n \geq 2$ , let $V$ be a complex vector space, and let $U_{1}, \dots, U_{n - 1} \in {End}_{C} (V)$ . Suppose $δ \in C$ , $q \in C^{\times}$ , and $δ = q + q^{- 1}$ , and assume

$U_{i}^{2} = δ U_{i} (1 \leq i < n), U_{i} U_{i + 1} U_{i} = U_{i}, U_{i + 1} U_{i} U_{i + 1} = U_{i + 1} (1 \leq i \leq n - 2), U_{i} U_{j} = U_{j} U_{i} (∣ i - j ∣ > 1) .$

Define $B_{i} : = q I_{V} - U_{i}$ . Then each $B_{i}$ is invertible with $B_{i}^{- 1} = q^{- 1} I_{V} - U_{i}$ ; the $B_{i}$ satisfy the Artin braid and far-commutation relations; and

$(B_{i} - q I_{V}) (B_{i} + q^{- 1} I_{V}) = 0.$

Thus $σ_{i} \mapsto B_{i}$ defines a group homomorphism $B_{n} \to GL (V)$ . If $H_{v} (n)$ is the generic type-A Hecke algebra with $(T_{i} - v) (T_{i} + 1) = 0$ , its scalar specialization $H_{q^{2}} (n)$ has a unital algebra representation $T_{i} \mapsto q B_{i}$ . This is the standard Hecke normalization because $(q B_{i} - q^{2} I_{V}) (q B_{i} + I_{V}) = 0$ . If $V$ is a complex Hilbert space and each $U_{i}$ is bounded self-adjoint, then $δ \in R$ and $∣ q ∣ = 1$ imply that each $B_{i}$ is unitary. No faithfulness, diagonalizability, or physical realization is asserted.

Statement references

Vacuum-channel anyonic projectors satisfy Temperley–Lieb relations

Vacuum projectors obey the reverse adjacent Temperley–Lieb relation

The generic type-A Hecke algebra

The Hecke generators satisfy the Artin relations and are units

The braid group by Artin presentation

Von Dyck's theorem: maps of generators that satisfy the relators extend uniquely from a presented group

Facts & Assumptions

Given: $n \geq 2$ , the operator family on $V$ , and $q, δ$ satisfying the hypotheses in the Statement.

[F1]

Vacuum-channel anyonic projectors satisfy Temperley–Lieb relations: on the repeated- $a$ path space $U_{i} = δ_{a} P_{i}$ with $δ_{a} = κ_{a} d_{a}$ ; its Statement gives the square relation, $U_{i} U_{i + 1} U_{i} = U_{i}$ , and distant commutation.

[F2]

Vacuum projectors obey the reverse adjacent Temperley–Lieb relation: on the same path space the omitted reverse relation $U_{i + 1} U_{i} U_{i + 1} = U_{i + 1}$ holds, so [F1] and [F2] supply the complete adjacent Temperley–Lieb relations.

[F3]

The generic type-A Hecke algebra: $H_{v} (n)$ is presented by generators $T_{i}$ , the type-A braid and distant-commutation relations, and $(T_{i} - v) (T_{i} + 1) = 0$ ; specialization at a unit $v = q^{2}$ preserves this presentation.

[F4]

The Hecke generators satisfy the Artin relations and are units: the generators of the Hecke tower are units and the standard assignment $σ_{i} \mapsto T_{i}$ defines the braid-group homomorphism into the Hecke unit group.

[F5]

The braid group by Artin presentation: $B_{n}$ has precisely the adjacent braid and distant-commutation relators displayed in the Statement.

[F6]

Von Dyck's theorem: maps of generators that satisfy the relators extend uniquely from a presented group: a map from the generators of a presented group extends uniquely to a homomorphism when every defining relator maps to the identity.

Proof

1.1givenalgebra

Since $U_{i}^{2} = δ U_{i}$ and $δ = q + q^{- 1}$ , $(q I_{V} - U_{i}) (q^{- 1} I_{V} - U_{i}) = I_{V} - (q + q^{- 1}) U_{i} + U_{i}^{2} = I_{V}$ . The product in the reverse order is the same polynomial in $U_{i}$ , so $B_{i}^{- 1} = q^{- 1} I_{V} - U_{i}$ .

1.2givenalgebra

For adjacent $i, i + 1$ , expansion and the square and adjacent Temperley–Lieb relations give $(q I_{V} - U_{i}) (q I_{V} - U_{i + 1}) (q I_{V} - U_{i}) - (q I_{V} - U_{i + 1}) (q I_{V} - U_{i}) (q I_{V} - U_{i + 1}) = - (q^{2} - q δ + 1) (U_{i} - U_{i + 1}) = 0$ , since $q δ = q^{2} + 1$ . Thus the adjacent braid relation holds.

2.1givenstep 1.1algebra

We have $B_{i} - q I_{V} = - U_{i}$ and $B_{i} + q^{- 1} I_{V} = δ I_{V} - U_{i}$ ; hence $(B_{i} - q I_{V}) (B_{i} + q^{- 1} I_{V}) = - δ U_{i} + U_{i}^{2} = 0$ .

2.2F5givenstep 1.1algebra

If $∣ i - j ∣ > 1$ , then $B_{i} B_{j} - B_{j} B_{i} = U_{i} U_{j} - U_{j} U_{i} = 0$ . Together with step 1.1, the $B_{i}$ are units satisfying all relators of [F5].

3.1F5F6step 2.2construct

By [F6] applied to [F5] and the assignment $σ_{i} \mapsto B_{i}$ , step 2.2 supplies a unique homomorphism $ρ_{n} : B_{n} \to GL (V)$ with $ρ_{n} (σ_{i}) = B_{i}$ .

3.2F3step 2.1step 1.2step 2.2algebra

Let ${\tilde{T}}_{i} : = q B_{i}$ . They satisfy the braid and distant relations by steps 1.2–2.2, and $({\tilde{T}}_{i} - q^{2} I_{V}) ({\tilde{T}}_{i} + I_{V}) = q^{2} (B_{i} - q I_{V}) (B_{i} + q^{- 1} I_{V}) = 0$ . Consequently the generator map from the free unital algebra kills every defining relator of $H_{q^{2}} (n)$ , so its defining ideal lies in the kernel and the map factors through the quotient $H_{q^{2}} (n)$ . This gives the stated unital Hecke-algebra representation.

4.1F3F4step 3.2construct

Apply the coefficient map $Z [v^{\pm 1}, z] \to C$ , $v \mapsto q^{2}$ , $z \mapsto 0$ , to the Hecke tower in [F4]. By its defining scalar-extension construction and the specialization in [F3], the resulting algebra is $H_{q^{2}} (n)$ ; base change carries [F4]'s braid homomorphism to $η_{n} : B_{n} \to H_{q^{2}} (n)^{\times}$ with $η_{n} (σ_{i}) = T_{i}$ . Composing with $Φ_{n}$ from step 3.2 sends $σ_{i}$ to $q B_{i}$ , identifying the chosen normalization against the standard Hecke generator assignment. No Markov trace or link-invariant claim is used.

5.1F1F2givenstep 1.1algebra∎

If $V$ is a complex Hilbert space, $U_{i}$ is bounded self-adjoint, $δ \in R$ , and $∣ q ∣ = 1$ , then for all $x, y \in V$ self-adjointness and $U_{i}^{2} = δ U_{i}$ give $⟨ B_{i} x, B_{i} y ⟩ = ∣ q ∣^{2} ⟨ x, y ⟩ - (q + \overline{q}) ⟨ x, U_{i} y ⟩ + δ ⟨ x, U_{i} y ⟩ = ⟨ x, y ⟩$ , since $q^{- 1} = \overline{q}$ and $δ = q + q^{- 1}$ . Thus $B_{i}$ preserves the inner product; it is surjective by its inverse in step 1.1, and so it is unitary by the convention in the Statement. For the repeated- $a$ anyonic path family, [F1] and [F2] supply both adjacent relations, the square relation, and distant commutation with the signed $δ_{a} = κ_{a} d_{a}$ ; whenever $q + q^{- 1} = δ_{a}$ , it therefore meets the operator hypotheses of this theorem. No sign is replaced by $+ 1$ .

Remarks

The finite-dimensional anyonic path-space operators are one algebraic source of the $U_{i}$ . The conclusion is conditional on their displayed relations and signed loop parameter. It does not identify these operators with an experimentally realized adiabatic exchange.

Temperley–Lieb projectors give a Hecke braid representation

Statement

Statement references

Facts & Assumptions

Proof

Remarks

Depends on

Dependency tree · 2 levels

Sources