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Temperley–Lieb Jones–Wenzl Projectors, Traces, and Quotients

1 · Prerequisites

2 · Summary

This page uses the supplier-first Jones–Wenzl, root-path and full rank-one tilting constructions on temperley-lieb-root-of-unity-and-modular-representation-theory. Their canonical carriers have a single home there; the full root-of-unity, fusion, trace-radical, centralizer and ribbon conclusions are retained here by reuse. The local result proves the exact Jones specialization of the Hecke Markov trace.

Results

The specialized Hecke Markov trace descends to the diagram quotient proves descent through the complete two-sided Hecke ideal, with $T_{i} \mapsto q U_{i} - 1$ and $z_{0} = - 1 / (q^{2} + 1)$ .
Unnormalized diagram closure and partial trace, Diagram closure trace is cyclic and satisfies the tower rules, Jones–Wenzl recursion over unit quantum integers and Jones–Wenzl projectors: annihilation, uniqueness, partial trace, and closure supply the exact unit-denominator recursion, unnormalized closure, projector uniqueness and partial traces.
General Temperley–Lieb theta-net evaluation and its root domain proves the general theta formula in the unit-denominator domain. At roots its retained labels have nonzero allowed-level pairings and zero forbidden-level pairings, with no undefined projector or boundary division.
Root fusion paths have nonzero pairings, Every boundary tensor-ideal component is the fixed embedded two-sided ideal and The root negligible quotient is semisimple with truncated fusion give all-rank matrix blocks, the embedded boundary ideal, the trace radical and truncated fusion. The unreduced first-boundary algebra is semisimple although its closure trace is degenerate.
The specified rank-one divided-power tilting category, Rank-one Weyl strings, tensor filtrations, and tilting classification, Outside-alcove rank-one tiltings have zero pivotal dimension and The rank-one tilting quotient is the truncated fusion category establish the full toral-character category, filtration and extension arguments, all indecomposable tiltings, endomorphism negligibility and fusion, for every $ℓ \geq 3$ .
The corrected finite rank-one R series gives the full tilting ribbon structure and The rank-one tilting quotient matches the Temperley-Lieb ribbon convention give the actual finite $R$ matrix, full semisimplified centralizer and signed-TL ribbon comparison. The self-dual cup/cap loop is $- δ$ ; parity changes the pivotal closure to $+ δ$ . The unreduced boundary projector has nonzero image and is killed only after the negligible quotient.

The trace-radical interface is Negligible morphisms for a pivotal trace and Partial-trace closure of negligible morphisms. Companion: temperley-lieb-jones-wenzl-traces-and-quotients-examples.

3 · Logical flowchart

4 · Definitions, theorems and proofs

TheoremStatement: Literature-sourcedProof: AI-adaptedOpen item page →

The specialized Hecke Markov trace descends to the diagram quotient

Statement

Let $ℓ \geq 3$ , $q = e^{π i / ℓ}$ , $v = q^{2}$ , $δ = q + q^{- 1}$ and $z_{0} = - 1 / (v + 1)$ . Specialize the type-A Hecke tower at $v, z = z_{0}$ . For each $n$ , let $J_{n}$ be the two-sided ideal generated by the six-term parabolic sums

$E_{i} = 1 + T_{i} + T_{i + 1} + T_{i} T_{i + 1} + T_{i + 1} T_{i} + T_{i} T_{i + 1} T_{i} (1 \leq i \leq n - 2) .$

The assignment $T_{i} \mapsto q U_{i} - 1$ defines a surjective tower map $ϕ_{n} : H (n) \to {T L}_{n} (C, δ)$ that kills $J_{n}$ . The specialized Ocneanu Markov trace therefore vanishes on $J_{n}$ and descends to $H (n) / J_{n}$ . Under the induced isomorphism with the diagram algebra presentation, the descended trace is the normalized closure trace $τ_{n} = δ^{- n} {Tr}_{n}$ .

Facts & Assumptions

Given: $ℓ \geq 3$ , $q = e^{π i / ℓ}$ , the Hecke parameter $v = q^{2}$ , the tower trace parameter $z_{0} = - 1 / (q^{2} + 1)$ , and the planar convention $U_{i}^{2} = δ U_{i}$ , $U_{i} U_{i + 1} U_{i} = U_{i}$ .

[F1]

The generic type-A Hecke generators obey $T_{i}^{2} = (v - 1) T_{i} + v$ , the braid relation, and distant commutation (The generic type-A Hecke algebra).

[F2]

$tr$ is cyclic, restricts along the tower, and satisfies ${tr}_{n + 1} (x T_{n}) = z {tr}_{n} (x)$ ; its recursive proof on the free tower basis works after any scalar specialization (The Markov trace on the type-A Hecke tower, The Ocneanu Markov trace exists and is unique).

[F3]

For $n \geq 0$ , the diagram closure has $τ_{n} (1_{n}) = 1$ and $τ_{n + 1} (x \otimes 1) = τ_{n} (x)$ . For $n \geq 1$ , it also has $τ_{n + 1} ((x \otimes 1) U_{n}) = δ^{- 1} τ_{n} (x)$ because $δ \neq 0$ (Unnormalized diagram closure and partial trace, Diagram closure trace is cyclic and satisfies the tower rules).

[F4]

The diagram algebra has the presentation on generators $U_{i}$ with relations $U_{i}^{2} = δ U_{i}$ , $U_{i} U_{i \pm 1} U_{i} = U_{i}$ , and distant commutation; its generator presentation is isomorphic to the planar diagram algebra over every commutative ring (The abstract Temperley–Lieb generator presentation, Elementary cup-cap diagrams generate every finite planar pairing endomorphism diagram, The generator presentation is the planar diagram algebra over every commutative ring).

[F5]

The Hecke tower is free over its preceding rank, with $H (n + 1) = H (n) \oplus H (n) T_{n} H (n)$ as an $H (n)$ -bimodule, and the standard Hecke basis persists after scalar specialization (The Hecke tower is free over the previous level, The standard basis of the generic type-A Hecke algebra).

Proof

1.1F1F4givenalgebra

Since $v + 1 = q^{2} + 1 = q δ \neq 0$ , the assignment $T_{i} \mapsto q U_{i} - 1$ is defined. Its quadratic relation follows from $(q U_{i} - 1 - v) (q U_{i}) = q^{2} (U_{i} - δ) U_{i} = 0$ , using $v = q^{2}$ and $U_{i}^{2} = δ U_{i}$ . For adjacent generators, expanding both braid words gives difference $q^{3} (U_{i} U_{i + 1} U_{i} - U_{i + 1} U_{i} U_{i + 1}) - q^{2} δ (U_{i} - U_{i + 1}) + q (U_{i} - U_{i + 1}) = 0$ , since $q^{2} δ = q^{3} + q$ . Distant commutation is inherited, so the assignment defines a tower homomorphism; it is surjective by [F4].

2.1F1step 1.1algebra

Put $E_{i} = \sum_{w \in ⟨ s_{i}, s_{i + 1} ⟩} T_{w}$ , the six-term expression in the Statement. Expanding with the Hecke quadratic and braid relations gives $(T_{i} + 1) (T_{i + 1} + 1) (T_{i} + 1) = E_{i} + v (T_{i} + 1)$ . Under $T_{j} + 1 = q U_{j}$ , the left side maps to $q^{3} U_{i} U_{i + 1} U_{i} = q^{3} U_{i}$ and the second term maps to $q^{3} U_{i}$ , so $ϕ_{n} (E_{i}) = 0$ . Since $J_{n}$ is the two-sided ideal generated by the $E_{i}$ , $ϕ_{n} (J_{n}) = 0$ and $ϕ_{n}$ factors through $H (n) / J_{n}$ .

2.2F2F3step 1.1algebra

The pullback ${\tilde{τ}}_{n} : = τ_{n} \circ ϕ_{n}$ is cyclic and has ${\tilde{τ}}_{n} (1) = 1$ by [F3]. For $n \geq 1$ and $x \in H (n)$ , its image in ${T L}_{n + 1}$ is $ϕ_{n} (x) \otimes 1$ , and $T_{n}$ maps to $q U_{n} - 1$ , so ${\tilde{τ}}_{n + 1} (x T_{n}) = (q / δ - 1) τ_{n} (ϕ_{n} (x)) = - {\tilde{τ}}_{n} (x) / (q δ) = - {\tilde{τ}}_{n} (x) / (q^{2} + 1) = z_{0} {\tilde{τ}}_{n} (x)$ . The tower restriction rule follows from [F3]; hence $\tilde{τ}$ satisfies the specialized Markov rules.

3.1F1F4step 1.1step 2.1algebra

The presentation [F4] gives $ψ_{n} : {T L}_{n} \to H (n) / J_{n}$ by $U_{i} \mapsto q^{- 1} (T_{i} + 1)$ . The Hecke quadratic relation gives $U_{i}^{2} = δ U_{i}$ for these images, while step 2.1 in the quotient gives $(T_{i} + 1) (T_{i + 1} + 1) (T_{i} + 1) = q^{2} (T_{i} + 1)$ and hence $U_{i} U_{i + 1} U_{i} = U_{i}$ ; the other adjacent relation follows by interchanging indices and distant commutation is inherited. Thus $ψ_{n}$ is well defined. Its composites with the map induced by $ϕ_{n}$ fix each generator: $U_{i} \mapsto q^{- 1} (q U_{i} - 1 + 1) = U_{i}$ and $T_{i} \mapsto q (q^{- 1} (T_{i} + 1)) - 1 = T_{i}$ . Therefore $H (n) / J_{n} ≅ {T L}_{n}$ , proving the quotient isomorphism.

4.1F2F5step 2.2step 3.1inductionalgebra∎

The decomposition in [F5] writes every element of $H (n + 1)$ as a sum of an element of $H (n)$ and terms $x T_{n} y$ with $x, y \in H (n)$ . For any Markov trace, restriction determines the first term; cyclicity and (M4) force ${tr}_{n + 1} (x T_{n} y) = z {tr}_{n} (x y)$ on each remaining term. Induction from $H (1) = C$ therefore shows at most one specialized trace can satisfy (M1)--(M4), even when $v, z$ have been specialized: this uses only the free decomposition [F5], not division by a parameter. By step 2.2 the pullback $\tilde{τ}$ is such a trace, so it is the specialized Ocneanu trace. Since $ϕ_{n} (J_{n}) = 0$ , the trace vanishes on all of the two-sided ideal $J_{n}$ . The quotient isomorphism of step 3.1 identifies the descended functional with the normalized closure trace $τ_{n}$ .

5 · Examples, counterexamples and false statements

None yet.