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Truncated fusion tables for ell four and ell five

Example

Apply the truncated rule of The rank-one tilting quotient is the truncated fusion category to list the full fusion tables for $ℓ = 4$ and $ℓ = 5$ . For $ℓ = 4$ the simple labels are $0, 1, 2$ and

$1 \otimes 1 = 0 \oplus 2, 1 \otimes 2 = 1, 2 \otimes 2 = 0.$

For $ℓ = 5$ the simple labels are $0, 1, 2, 3$ and

$\begin{aligned} 1 \otimes 1 & = 0 \oplus 2, & 1 \otimes 2 & = 1 \oplus 3, & 1 \otimes 3 & = 2, \\ 2 \otimes 2 & = 0 \oplus 2, & 2 \otimes 3 & = 1, & 3 \otimes 3 & = 0. \end{aligned}$

In both tables $0$ is the tensor unit and the products with $0$ are $0 \otimes a = a$ .

Facts & Assumptions

Given: The truncated fusion rule and the retained simple-label ranges for each root order.

[F1]

At order $ℓ$ , the retained labels are $0, 1, \dots, ℓ - 2$ and $a \otimes b$ contains $c = ∣ a - b ∣, ∣ a - b ∣ + 2, \dots, \min (a + b, 2 ℓ - 4 - a - b)$ (The rank-one tilting quotient is the truncated fusion category, The root negligible quotient is semisimple with truncated fusion).

[F2]

The conversion theorem identifies the pivotal/ribbon rank-one labels with the same retained Temperley–Lieb fusion labels (The rank-one tilting quotient matches the Temperley-Lieb ribbon convention).

Verification

technique · direct calculation

1.1F1givenalgebra

For $ℓ = 4$ , the cutoff is $2 ℓ - 4 = 4$ . Substituting $(a, b) = (1, 1), (1, 2), (2, 2)$ into [F1] gives upper bounds $\min (2, 2) = 2$ , $\min (3, 1) = 1$ , and $\min (4, 0) = 0$ , respectively; the lower bounds and parity give exactly ${0, 2}$ , ${1}$ , and ${0}$ . The unit rule supplies products with $0$ .

2.1F1F2step 1.1algebra∎

For $ℓ = 5$ , the cutoff is $6$ . The substitutions $(1, 1), (1, 2), (1, 3), (2, 2), (2, 3), (3, 3)$ give intervals and parity sets ${0, 2}$ , ${1, 3}$ , ${2}$ , ${0, 2}$ , ${1}$ , and ${0}$ , respectively. These exhaust unordered pairs among $0, 1, 2, 3$ ; commutativity of the fusion rule supplies the reversed pairs. The conversion in [F2] preserves the labels and cutoff.

Depends on

Used by

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Sources

K. Iohara, G. I. Lehrer, and R. B. Zhang, The Jones Quotients of the Temperley-Lieb Algebras (standard reference, not scraped)
Stephen F. Sawin, Quantum Groups at Roots of Unity and Modularity (standard reference, not scraped)