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Temperley–Lieb Root-of-Unity and Modular Representation Theory: Examples and Counterexamples

1 · Prerequisites

2 · Summary

These examples check one noncritical Gram radical, one critical simple standard module, and one Spencer digit-support calculation. Each verification states its exact parameter and label convention.

The first example’s TikZ half-diagrams specify the two noncrossing pairings. Their labels are the mathematical objects; the loop-count proof computes the Gram entries independently of the picture.

Examples

An order-three noncritical link-state form with a one-dimensional radical computes the 2-by-2 Gram matrix at rank $n = 4$ and its one-dimensional radical at $q^{2}$ of order three; its label is noncritical.
A critical five-defect standard module at order three is simple checks a rank-seven, five-defect critical standard module and its nonzero determinant at the same root.
The Spencer support set for rank label eleven in characteristic three expands $12$ in the $(ℓ, P) = (3, 3)$ scales and obtains ${11, 5}$ ; it makes no composition-factor claim without Spencer’s separate theorem.

The first two examples illustrate the complex standard-module classification on the companion page. The arithmetic support calculation makes no independent Spencer decomposition claim, and these examples supply no TLJ fusion or tilting-centralizer theorem.

Companion: temperley-lieb-root-of-unity-and-modular-representation-theory.

3 · Logical flowchart

4 · Definitions, theorems and proofs

None yet.

5 · Examples, counterexamples and false statements

ExampleConstruction: Literature-sourcedVerification: AI-adaptedjudge pass (gpt-6-luna)Open item page →

An order-three noncritical link-state form with a one-dimensional radical

Example

Take $q = e^{π i / 3}$ , so $δ = 1$ and $κ = ord (q^{2}) = 3$ . At rank $n = 4$ with $p = 2$ arcs and $t = 0$ defects, the link-state Gram matrix has a one-dimensional radical even though this label is noncritical.

Diagram

The sites are labeled from left to right. In each half-diagram, an arc joins its two labeled sites; the left panel is $x = (12) (34)$ and the right panel is $y = (14) (23)$ . These are the two noncrossing perfect matchings on four sites; the picture records their basis labels, while the Gram entries below are computed from the overlay loop count.

\begin{tikzpicture}[scale=0.9]
  \foreach \x in {0,1,2,3} { \fill (\x,0) circle (1.5pt); }
  \node[below] at (0,-0.05) {$1$};
  \node[below] at (1,-0.05) {$2$};
  \node[below] at (2,-0.05) {$3$};
  \node[below] at (3,-0.05) {$4$};
  \draw (0,0) .. controls (0,0.7) and (1,0.7) .. (1,0);
  \draw (2,0) .. controls (2,0.7) and (3,0.7) .. (3,0);
  \node at (1.5,1.0) {$x=(12)(34)$};

  \foreach \x in {5,6,7,8} { \fill (\x,0) circle (1.5pt); }
  \node[below] at (5,-0.05) {$1$};
  \node[below] at (6,-0.05) {$2$};
  \node[below] at (7,-0.05) {$3$};
  \node[below] at (8,-0.05) {$4$};
  \draw (5,0) .. controls (5,1.45) and (8,1.45) .. (8,0);
  \draw (6,0) .. controls (6,0.7) and (7,0.7) .. (7,0);
  \node at (6.5,1.7) {$y=(14)(23)$};
\end{tikzpicture}

Facts & Assumptions

Given: $q = e^{π i / 3}$ , $n = 4$ , $p = 2$ , and the link-state capping form.

[F1]

The standard-module basis is the set of noncrossing partial matchings with the indicated number of arcs and defects (Temperley–Lieb link-state standard modules by defect number).

[F2]

The form value on two basis states is $δ^{c}$ when all defect paths propagate, with $c$ the number of overlay loops (The link-state Gram form is symmetric and invariant).

[F3]

The Gram determinant is the product of quantum-integer ratios with polynomial continuation at removable zero-over-zero factors (The Temperley–Lieb link-state Gram determinant).

[F4]

A label is critical exactly when $κ < \infty$ and $κ ∣ (t + 1)$ (Complex Temperley–Lieb critical labels and reflection).

Verification

Given: $δ = 1$ , $κ = 3$ , and the two perfect matchings $x, y$ shown above.

1.1F1F2givenconstruct

There are exactly two noncrossing perfect matchings on four ordered sites: $x = (12) (34)$ and $y = (14) (23)$ . Overlaying either matching with itself gives two loops and hence Gram value $δ^{2} = 1$ ; overlaying $x$ with $y$ gives one loop and hence Gram value $δ = 1$ . Thus the matrix in the ordered basis $(x, y)$ is $G_{4, 2} = (\begin{array}{cc} 1 & 1 \\ 1 & 1 \end{array}) .$

2.1step 1.1algebra

A vector $a x + b y$ is orthogonal to both $x$ and $y$ exactly when $a + b = 0$ , by the two equal rows of $G_{4, 2}$ . Therefore $rad ϕ_{4, 0} = C (x - y)$ , a one-dimensional subspace.

3.1F3F4givenstep 1.1algebra∎

Here $t = n - 2 p = 0$ , so $t + 1 = 1$ is not divisible by $κ = 3$ ; the label is noncritical by [F4]. The determinant formula [F3] gives $\det G_{4, 2} = {(\frac{[2]}{[1]})}^{3} (\frac{[3]}{[2]}) = 0,$ because $[1] = [2] = 1$ and $[3] = 0$ at $q = e^{π i / 3}$ . This agrees with the explicitly computed radical.

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A critical five-defect standard module at order three is simple

Example

Let $q = e^{π i / 3}$ , so $δ = 1$ and $κ = 3$ . At rank $n = 7$ , take $p = 1$ arc and $t = 5$ defects. The critical standard module is simple, and its Gram determinant is $1$ .

Facts & Assumptions

Given: $q = e^{π i / 3}$ , $n = 7$ , $p = 1$ , $t = 5$ , and $κ = 3$ .

[F1]

The critical-label test is $κ ∣ (t + 1)$ (Complex Temperley–Lieb critical labels and reflection).

[F2]

At every critical label the complex link-state capping form is nondegenerate and its standard module is simple (Critical complex Temperley–Lieb standard modules are simple).

[F3]

For $p = 1$ , the Gram determinant is $[t + 2]_{q} / [1]_{q}$ (The Temperley–Lieb link-state Gram determinant).

Verification

Given: $δ = 1$ , $κ = 3$ , and the rank-seven label $(n, p) = (7, 1)$ .

1.1F1givenalgebra

The defect number is $t = 7 - 2 = 5$ , so $t + 1 = 6$ is divisible by $κ = 3$ . Hence $(7, 1)$ is critical by [F1].

2.1F2F3givenstep 1.1algebra∎

Since $p = 1$ , [F3] gives $\det G_{7, 1} = [7]_{q} / [1]_{q}$ . Here $[1]_{q} = 1$ and $[7]_{q} = \frac{\sin (7 π / 3)}{\sin (π / 3)} = 1,$ so $\det G_{7, 1} = 1 \neq 0$ . By [F2], this nondegenerate critical standard module is simple.

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The Spencer support set for rank label eleven in characteristic three

Example

In a field of characteristic $3$ with loop parameter $δ = 2$ , the recurrence quantum characteristic is $ℓ = 3$ . For rank label $N = 11$ , Spencer's support set is ${11, 5}$ .

Facts & Assumptions

Given: $char K = 3$ , $δ = 2 \in K$ , $ℓ = 3$ , and $N = 11$ .

[F1]

The recurrence is $[0] = 0$ , $[1] = 1$ , $[r + 1] = δ [r] - [r - 1]$ , and $ℓ$ is the least positive index with $[ℓ] = 0$ (Temperley–Lieb quantum integers and quantum characteristic).

[F2]

If $ℓ < \infty$ , the digits are those of $N + 1$ in the scales $s_{0} = 1$ , $s_{i} = ℓ P^{i - 1}$ , and the support set uses a positive top sign and arbitrary lower signs (Spencer quantum-characteristic digits and support set).

Verification

Given: $K$ has characteristic $3$ , $δ = 2$ , and $N = 11$ .

1.1F1givenalgebra

In $K$ , $[1] = 1$ , $[2] = 2$ , and $[3] = 2 \cdot 2 - 1 = 3 = 0$ , so the least positive vanishing index is $ℓ = 3$ . Here $P = 3$ , and the scales are $s_{0} = 1$ , $s_{1} = 3$ , $s_{2} = 9$ .

2.1F2givenstep 1.1algebra

The digits of $N + 1 = 12$ are $N_{0} = 0$ , $N_{1} = 1$ , $N_{2} = 1$ , because $12 = 0 \cdot 1 + 1 \cdot 3 + 1 \cdot 9$ . With highest index $2$ , the support values are $1 \cdot 9 \pm 1 \cdot 3 \pm 0 \cdot 1 - 1$ , namely $11$ and $5$ . Both are nonnegative, so ${Supp}_{3, 3} (11) = {11, 5}$ .

3.1F2step 2.1∎

This computation determines only the set in the definition; identifying its elements with actual composition factors requires the separate Spencer decomposition theorem and is not asserted by this example.

Temperley–Lieb Root-of-Unity and Modular Representation Theory: Examples and Counterexamples

1 · Prerequisites

2 · Summary

Examples

3 · Logical flowchart

4 · Definitions, theorems and proofs

5 · Examples, counterexamples and false statements

An order-three noncritical link-state form with a one-dimensional radical

Example

Diagram

Facts & Assumptions

Verification

A critical five-defect standard module at order three is simple

Example

Facts & Assumptions

Verification

The Spencer support set for rank label eleven in characteristic three

Example

Facts & Assumptions

Verification

Sources