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Potts local algebra and braid quotient

Statement

Let $Q \in {2, 3, 4}$ , $L \geq 1$ , $δ : = \sqrt{Q}$ , and let $q \in C ∖ {0}$ satisfy $q + q^{- 1} = δ$ . Use the $2 L - 1$ open-chain operators $e_{i}$ defined for this Potts-clock chain, and put $g_{i} : = q I - e_{i}$ for $1 \leq i \leq 2 L - 1$ . Then $u_{i} \mapsto e_{i}$ represents the rank- $2 L$ Temperley-Lieb algebra $P_{2 L} (C, δ)$ , and $(g_{i} - q I) (g_{i} + q^{- 1} I) = 0, g_{i} g_{i + 1} g_{i} = g_{i + 1} g_{i} g_{i + 1}, g_{i} g_{j} = g_{j} g_{i} (∣ i - j ∣ > 1) .$ These are the type-A Hecke relations in the normalized generator convention after the coefficient specialization $v \mapsto q^{- 1}$ . The specified clock-shift model selects $q = e^{i π / 4}, e^{i π / 6}, 1$ for $Q = 2, 3, 4$ , respectively. At $Q = 4$ , $q = 1$ and the same operators $g_{i} = I - e_{i}$ satisfy the symmetric-group Coxeter relations, so they define an action of $S_{2 L}$ . No faithfulness, semisimplicity, or unitarity is asserted.

Statement references

Finite Hermitian Potts-clock chain

Open Potts clock-shift Temperley–Lieb generators

The abstract Temperley–Lieb generator presentation

The normalized type-A Hecke algebra and its bar involution

The symmetric group has the Coxeter presentation

Facts & Assumptions

Given: $Q \in {2, 3, 4}$ , integer $L \geq 1$ , $δ = \sqrt{Q}$ , and $q \in C ∖ {0}$ with $q + q^{- 1} = δ$ .

[A1]

The parameter and chain-length hypotheses are exactly those in the Statement.

[F1]

Finite Hermitian Potts-clock chain and Open Potts clock-shift Temperley–Lieb generators: the space is $H_{Q, L} = (C^{Q})^{\otimes L}$ ; the local operators are $e_{2 j - 1} = \sqrt{Q} P_{j}$ and $e_{2 j} = \sqrt{Q} D_{j, j + 1}$ , equivalently the cyclic clock-shift sums; the open endpoints have $2 L - 1$ generators, and the selected $q$ values are $e^{i π / 4}, e^{i π / 6}, 1$ for $Q = 2, 3, 4$ .

[F2]

The Q = 2, 3, 4 open Potts matrices satisfy Temperley–Lieb: for each $Q \in {2, 3, 4}$ , $e_{i}^{2} = \sqrt{Q} e_{i}$ , $e_{i} e_{i + 1} e_{i} = e_{i} = e_{i + 1} e_{i} e_{i + 1}$ , and $e_{i} e_{j} = e_{j} e_{i}$ when $∣ i - j ∣ > 1$ ; its chosen $q$ values satisfy $\sqrt{Q} = q + q^{- 1}$ .

[F3]

The abstract Temperley–Lieb generator presentation: $P_{2 L} (C, δ)$ has generators $u_{1}, \dots, u_{2 L - 1}$ with exactly the quadratic, adjacent triple-product, and distant-commutation relations in [F2].

[F4]

The normalized type-A Hecke algebra and its bar involution: over $Z [v^{\pm 1}]$ , its generator satisfies $H_{i}^{2} = 1 + (v^{- 1} - v) H_{i}$ and the braid and distant-commutation relations; after the coefficient map $v \mapsto q^{- 1}$ this quadratic is $(H_{i} - q) (H_{i} + q^{- 1}) = 0$ .

[F5]

The symmetric group has the Coxeter presentation: the adjacent transpositions of $S_{2 L}$ have relations $s_{i}^{2} = 1$ , $s_{i} s_{i + 1} s_{i} = s_{i + 1} s_{i} s_{i + 1}$ , and distant commutation, and these relations present $S_{2 L}$ .

Proof

1.1A1F2F3algebra

By [F2], every defining relation of $P_{2 L} (C, δ)$ is satisfied by the operators $e_{i}$ ; the quotient presentation in [F3] therefore gives the stated Temperley-Lieb representation.

1.2A1F2F4algebra

For each $i$ , $g_{i} - q I = - e_{i}$ and $g_{i} + q^{- 1} I = δ I - e_{i}$ . Hence $(g_{i} - q I) (g_{i} + q^{- 1} I) = - e_{i} (δ I - e_{i}) = e_{i}^{2} - δ e_{i} = 0$ by [F2] and $δ = q + q^{- 1}$ . This is the Hecke quadratic relation in [F4]'s normalization.

1.3A1F2algebra

For adjacent $i, i + 1$ , expand with $e = e_{i}$ and $f = e_{i + 1}$ . Using $e^{2} = δ e$ , $f^{2} = δ f$ , $e f e = e$ , and $f e f = f$ from [F2], direct multiplication gives $g_{i} g_{i + 1} g_{i} - g_{i + 1} g_{i} g_{i + 1} = (- q^{2} + q δ - 1) (e_{i} - e_{i + 1}) .$ Since [A1] gives $q δ = q (q + q^{- 1}) = q^{2} + 1$ , this difference is zero.

1.4F2algebra

If $∣ i - j ∣ > 1$ , [F2] gives $e_{i} e_{j} = e_{j} e_{i}$ , so their affine combinations $g_{i} = q I - e_{i}$ and $g_{j} = q I - e_{j}$ commute.

2.1F1F2F5step 1.3step 1.4algebra∎

When $Q = 4$ , [F1] selects $q = 1$ and $δ = 2$ . Then $g_{i} = I - e_{i}$ and $g_{i}^{2} = (I - e_{i})^{2} = I - 2 e_{i} + e_{i}^{2} = I$ by [F2]. Steps 1.3–1.4 give the adjacent braid and distant-commutation relations. These are exactly [F5]'s Coxeter presentation, so the assignment of adjacent transpositions of $S_{2 L}$ to the $g_{i}$ defines a group action. This does not assert that the action is faithful.

Remarks

The local matrices $e_{i}$ are Hermitian projectors scaled by $\sqrt{Q}$ , but this does not make $g_{i} = q I - e_{i}$ a star representation when $q$ is non-real. The $Q = 2, 3$ braid and Hecke relations above are algebraic statements only; no semisimplicity, faithfulness, integrability, or unitary deformation follows from them. The $Q = 4$ symmetric-group action is a separate finite representation, not a claim about the dimension or equivalence of any other Potts or spin representation.

The first diagram uses the Temperley-Lieb rectangle convention: boundary points are read left to right, composition stacks bottom to top, and each closed loop is assigned the scalar $δ$ . The second diagram reads braid words bottom to top. At each crossing, the right-to-left strand is the over-strand: its white underlay opens a gap in the other strand, then its black centerline is drawn across that gap. Both diagrams illustrate conventions only; the proofs above are algebraic.

Potts local algebra and braid quotient

Statement

Statement references

Facts & Assumptions

Proof

Remarks

Depends on

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