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Fusion multiplicity space

Definition

Let $a, b, c$ be simple objects of a unitary fusion category $C$ (Unitary fusion category). The fusion multiplicity is

$N_{a b}^{c} : = \dim_{C} {Hom}_{C} (a \otimes b, c),$

and the corresponding fusion multiplicity space is

$V_{a b}^{c} : = {Hom}_{C} (a \otimes b, c) .$

Since $c$ is absolutely simple (Absolutely simple objects), each $f, g \in V_{a b}^{c}$ has a unique scalar $⟨ f, g ⟩$ satisfying $f \circ g^{†} = ⟨ f, g ⟩ {id}_{c}$ . This fixes the inner-product convention to be linear in the first variable. The next local theorem proves that this pairing is positive definite. No multiplicity-free assumption is made: $N_{a b}^{c}$ may be any nonnegative integer.

Depends on

Used by

Fusion-basis gauge change Definition
Multiplicity-aware fusion-tree basis Definition
R-symbol as a channel exchange map Definition
Fibonacci fusion data and chiral convention Example
Ising fusion data and chiral convention Example
All-label unitary Ising normal data and ribbon coherence Lemma
Fusion-tree basis and dimension recursion Lemma
Fusion multiplicity spaces are finite-dimensional inner-product spaces Theorem
Periodic Ising paths, two sectors, and the full line representation Theorem
Unitary fusion paths form an orthonormal Hilbert basis Theorem

Dependency tree · two levels

7 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

David Penneys, Unitary Fusion Categories, Math 8800 lecture notes (standard reference, not scraped)
Chetan Nayak et al., Non-Abelian Anyons and Topological Quantum Computation (standard reference, not scraped)