LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generated

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Distinct fusion-ring characters are orthogonal under duality

Statement

Let $A$ be a fusion ring over $Z$ with finite distinguished basis $I$ , unit $0$ , and duality $i \mapsto i^{*}$ . If $χ, ψ : A \to C$ are distinct unital ring homomorphisms, then $\sum_{i \in I} χ (i) ψ (i^{*}) = 0$ .

Facts & Assumptions

Given: A finite fusion ring $A$ with its distinguished basis, duality, and two distinct unital ring characters $χ, ψ$ .

[F1]

Multiplication has structure constants $N_{a b}^{c}$ in the distinguished basis, and Frobenius reciprocity gives $N_{y, x^{*}}^{z} = N_{y^{*}, z}^{x^{*}}$ (Fusion rules, Fusion and multifusion categories).

[F2]

A unital ring character is additive, multiplicative, and sends the unit basis element to $1$ .

Proof

Given: The data in the Statement.

1.1givenF1F2algebra

For a basis element $y$ , form $P_{χ} = \sum_{x \in I} χ (x) x^{*}$ in $A \otimes_{Z} C$ . The coefficient of $z$ in $y P_{χ}$ is $\sum_{x} χ (x) N_{y, x^{*}}^{z} = \sum_{u} χ (u^{*}) N_{y^{*}, z}^{u}$ by [F1], after setting $u = x^{*}$ . By multiplicativity and the anti-involution identity $(y^{*} z)^{*} = z^{*} y$ , this equals $χ (z^{*}) χ (y)$ , which is the coefficient of $z$ in $χ (y) P_{χ}$ . Hence $y P_{χ} = χ (y) P_{χ}$ .

2.1givenstep 1.1F2algebra∎

Applying $ψ$ to the equality in step 1.1 gives $ψ (y) ψ (P_{χ}) = χ (y) ψ (P_{χ})$ , where $ψ (P_{χ}) = \sum_{x} χ (x) ψ (x^{*})$ . Since $χ \neq ψ$ , some $y \in A$ has $χ (y) \neq ψ (y)$ ; subtracting the two sides forces $ψ (P_{χ}) = 0$ . This is the asserted sum.

Depends on

Used by

The modular Hopf-link matrix squares to D² times duality Lemma

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

Etingof, Gelaki, Nikshych, and Ostrik, Tensor Categories, Lemma 8.14.1 (standard reference, not scraped)
Etingof, Nikshych, and Ostrik, On Fusion Categories, §2.1 (standard reference, not scraped)