SU(2) Conventions

This section documents the mathematical conventions used by the SU(2) reference implementation of the Yuzuha Protocol. The protocol requires each implementation to adopt a self-consistent convention, but does not mandate any specific choice. The conventions below represent one valid option; alternative implementations are free to use different phase choices, fusion-tree coupling orders, or OM-index labelings provided they remain internally consistent and their APIs satisfy the Symbol Interface and Duality Interface.

Structure

The conventions are divided into three pages, each covering a logically distinct aspect of the SU(2) implementation:

Page	Contents
Fusion Conventions	Left-associative fusion tree; lexicographic OM-index ordering
Basis Conventions	Condon-Shortley CG coefficients; orthonormal basis normalization; canonical edge directions
Arrow Conventions	Charge-conservation sign rule; invariant metric; arrow reversal; Frobenius-Schur indicator; canonical conjugate pattern

Fusion Conventions

The SU(2) reference implementation uses a left-associative fusion tree. For an \(n\)th-order tensor with external representations \((j_1, \ldots, j_n)\), the internal representation sequence \((\alpha_1, \ldots, \alpha_{n-2})\) is built iteratively from the left:

\[\bigl(\cdots\bigl((j_1 \otimes j_2) \otimes j_3\bigr) \otimes \cdots\bigr) \otimes j_n\]

Valid internal sequences are enumerated in lexicographic order, with the OM index \(\mu = 0\) assigned to the smallest sequence. See Fusion Conventions for the full specification.

Basis Conventions

Clebsch-Gordan coefficients follow the Condon-Shortley phase convention:

\[C^{j_3 m_3}_{j_1 m_1,\, j_2 m_2} = \langle j_1 m_1,\, j_2 m_2 \mid j_3 m_3 \rangle\]

Their values are computed using the Racah formula, which expresses each CG coefficient as an explicit algebraic sum over factorials of angular momentum quantum numbers.

The canonical basis tensors \(\{B_\mu\}\) form an orthonormal set over the CG tensor space. For a 3rd-order tensor, the single basis element is normalized by \(1/\sqrt{2j_3 + 1}\) relative to the bare CG coefficient.

Each basis tensor is computed in a fixed canonical direction: all leading edges (first \(n-1\) edges) are incoming and the terminal edge (last edge) is outgoing. Bases with other direction combinations are derived by contracting deviating edges with the invariant metric \(g^{(j)}\). See Basis Conventions for the full specification.

Arrow Conventions

The Direction type assigns \(+1\) to Incoming and \(-1\) to Outgoing at the protocol level (see Type System — Direction). The signed magnetic-number sum must therefore vanish for any non-zero basis element:

\[\sum_{\text{incoming}} m_i - \sum_{\text{outgoing}} m_j = 0\]

The invariant metric \(g^{(j)}\) mediates between a representation and its dual:

\[g^{(j)}_{m,\,m'} = (-1)^{j-m}\,\delta_{m,\,-m'}\]

It satisfies \((g^{(j)})^2 = (-1)^{2j}\,I\), so its square equals the Frobenius-Schur (FS) indicator \(\text{FS}(j) = (-1)^{2j}\). The FS indicator is \(+1\) for integer spin and \(-1\) for half-integer spin; it sets the phase acquired per bond inversion. See Arrow Conventions for the full specification.