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Critical complex Temperley–Lieb standard modules are simple

Statement

Let $q \in C^{\times}$ , $δ = q + q^{- 1}$ , $n \geq 0$ , and $0 \leq p \leq ⌊ n / 2 ⌋$ . Put $t = n - 2 p$ , let $κ = ord (q^{2})$ , and let $ϕ_{n, t}$ be the link-state capping form on $W_{n, t} (C, δ)$ . If $(n, p)$ is critical in the sense of Complex Temperley–Lieb critical labels and reflection, then $ϕ_{n, t}$ is nondegenerate and $W_{n, t} (C, δ)$ is simple.

Facts & Assumptions

Given: $q \in C^{\times}$ , $δ = q + q^{- 1}$ , an allowed $(n, p)$ , and the critical-label convention.

[F1]

The complex quantum integers satisfy $[r]_{q} = (q^{r} - q^{- r}) / (q - q^{- 1})$ for $q \neq \pm 1$ , and $[r]_{q} = r q^{r - 1}$ for $q = \pm 1$ ; the recurrence quantum characteristic is distinct from $κ$ at $q = \pm 1$ (Temperley–Lieb quantum integers and quantum characteristic).

[F2]

$W_{n, t}$ has the link-state basis indexed by $p$ -arc partial matchings, and $ϕ_{n, t}$ is the reflection-and-capping form (Temperley–Lieb link-state standard modules by defect number, The link-state Gram form is symmetric and invariant).

[F3]

For $d_{n, r} = (\binom{n}{r}) - (\binom{n}{r - 1})$ with $(\binom{n}{- 1}) = 0$ , its Gram determinant is $\det G_{n, p} = \prod_{j = 1}^{p} {(\frac{[t + 1 + j]_{q}}{[j]_{q}})}^{d_{n, p - j}}$ , where the entire product denotes its polynomial continuation in $δ$ ; individual quotients are initially rational functions (The Temperley–Lieb link-state Gram determinant).

[F4]

The nonzero cell-form quotient is simple; when the cell form is nondegenerate its radical is zero, so the quotient is the entire standard module (Nonzero Temperley–Lieb cell forms give all simple heads).

[F5]

Criticality means $κ < \infty$ and $κ ∣ (t + 1)$ ; at $κ = 1$ this includes every allowed label (Complex Temperley–Lieb critical labels and reflection).

Proof

technique · Evaluate each determinant factor, including its removable root limit

1.1F1F3F5givenalgebra

If $κ = 1$ , then $q = \pm 1$ . By [F1], every positive quantum integer is nonzero, so each factor $[t + 1 + j]_{q} / [j]_{q}$ in [F3] is nonzero. Therefore $\det G_{n, p} \neq 0$ in this case.

1.2F1F3F5givenalgebra

Suppose $κ \geq 2$ . By [F5], there is an integer $m \geq 1$ with $t + 1 = m κ$ . Fix a root $q_{0} = q$ and write $ϵ = q_{0}^{m κ} \in {1, - 1}$ . For any $j$ not divisible by $κ$ , $[j]_{q_{0}} \neq 0$ and $[m κ + j]_{q_{0}} = ϵ [j]_{q_{0}}$ , by substituting $q_{0}^{m κ} = q_{0}^{- m κ} = ϵ$ in the quantum-integer formula. Thus this quotient is nonzero.

2.1F1F3step 1.2algebra

If $j = s κ$ with $s \geq 1$ , the quotient in [F3] has a removable zero-over-zero. Write nearby parameters as $q = e^{u}$ , with $u_{0}$ satisfying $e^{κ u_{0}} = ϵ_{0} \in {1, - 1}$ . Since $κ \geq 2$ , $q_{0} \neq \pm 1$ and $\sinh u_{0} \neq 0$ . From $[a κ]_{e^{u}} = \sinh (a κ u) / \sinh u$ , the ratio has limit $\lim_{u \to u_{0}} \frac{[(m + s) κ]_{e^{u}}}{[s κ]_{e^{u}}} = \frac{m + s}{s} (- 1)^{m v} \neq 0$ , where $κ u_{0} = π i v$ for an integer $v$ ; this follows by taking the ratio of the first derivatives of the two vanishing numerators. Also $d (q + q^{- 1}) / d q = 1 - q^{- 2} \neq 0$ at $q_{0}$ , so this quotient also has a removable continuation locally in $δ$ at the chosen root. Thus every rational factor in [F3] is regular and nonzero there, and their product equals the determinant polynomial by [F3].

3.1F2F3step 1.1step 2.1algebra

The exponents $d_{n, p - j}$ in [F3] are nonnegative integers, and an empty product is $1$ . Steps 1.1–2.1 therefore give $\det G_{n, p} \neq 0$ for every critical label, so the finite-dimensional form $ϕ_{n, t}$ is nondegenerate.

4.1F2F4step 3.1∎

By [F4], a nondegenerate cell form has zero radical and its quotient by that radical is simple. The quotient is $W_{n, t}$ itself, so this standard module is simple.

Depends on

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Sources

David Ridout and Yvan Saint-Aubin, Standard Modules, Induction and the Structure of the Temperley-Lieb Algebra (standard reference, not scraped)
Andrew Mathas, Hecke algebras and Schur algebras of the symmetric group, complete prepublication manuscript (standard reference, not scraped)