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Fusion Conventions

Left-Associative Fusion Trees

For an \(n\)th-order tensor with external representations \((j_1, j_2, \ldots, j_n)\), the canonical fusion tree is left-associative:

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

This uniquely determines the structure of the internal representation sequence \((\alpha_1, \ldots, \alpha_{n-2})\):

  • \(\alpha_1\) is an admissible representation in \(j_1 \otimes j_2\)
  • \(\alpha_2\) is an admissible representation in \(\alpha_1 \otimes j_3\)
  • \(\vdots\)
  • \(\alpha_{n-2}\) is an admissible representation in \(\alpha_{n-3} \otimes j_{n-1}\), and must equal \(j_n\) (fixed by the terminal edge)

Any alternative fusion-tree ordering (e.g. right-associative, binary-tree) is not conformant with this convention and will produce incompatible symbols.

Canonical Ordering of Internal Representations

For a given set of external edges, the valid internal representation sequences are enumerated in lexicographic order of the sequence \((\alpha_1, \ldots, \alpha_{n-2})\), where each \(\alpha_k\) is compared using the total order defined by the RepLabel type. The OM index \(\mu\) maps to this ordering: \(\mu = 0\) is the lexicographically smallest valid sequence.