Categorifiability Criteria

In the criteria on this page we use the following notation:

• \(R\): a fusion ring with basis \(\left(b_i\right)\) and structure constants \(N_{ij}^k\)
• \(X_i\): the fusion matrices, i.e. \([X_i]^k_j = N_{ij}^k\)
• \(A := \sum_i X_i X_i^*\) (with \(X_i^* := X_{i^*}\)), and with eigenvalues \((c_j)\). These are also called the formal codegrees of a fusion ring.
• \(\lambda\): the matrix that simultaneously diagonalizes all \(X_i\) (if it exists), i.e. \([\lambda]^i_j\) are the characters of \(R\).

Criteria for General Categorification

Criteria for Complex Categorification

d-number criterion

Definition (\(d\)-number) An algebraic integer \(\alpha\) is called a \(d\)-number if its minimal polynomial \(p(x) = x^n + a_1x^n-1+\cdots+a_n\) (where \(a_i \in \mathbb{Z}\)) satsifies that \((a_n)^i\) divides \((a_i)^n\) for all \(i\).

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Theorem (\(d\)-number criterion) Let \(R\) be commutative. If \(R\) admits a complex fusion category, then the formal codegrees \((c_j)\) of \(R\) are \(d\)-numbers.

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Extended Cyclotomic Criterion

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Theorem (Extended Cyclotomic Criterion) Let \(R\) be commutative. If there is a fusion matrix such that the splitting field of its minimal polynomial is a non-abelian extension of \(\mathbb{Q}\) then \(R\) admits no complex categorification.

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Criteria for Pivotal Categorification

Pivotal Version of Drinfeld Center Criterion

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Theorem (Pivotal version of Drinfeld center criterion) Let \(R\) be commutative. If \(R\) admits a complex pivotal categorification, then there exists \(j\) such that for all \(i, c_j / c_i\) is an algebraic integer.

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Criteria for Unitary Categorification

Schur Product Criterion

The commutative Schur product criterion (corollary 8.5) is the following:

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Theorem (commutative Schur product criterion) Let \(R\) be commutative with \([\lambda]^i_1=\max _j\left(\left|\lambda_{i, j}\right|\right)\). If \(R\) admits a unitary categorification, then for all triples \(\left(j_1, j_2, j_3\right)\) we have

\[\sum_i \frac{ [\lambda]^i_{j_1} [\lambda]^i_{j_2} [\lambda]^i_{j_3} }{[\lambda]^i_1} \geq 0\]

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Note that Theorem is the corollary of a (less tractable) noncommutative version (Proposition 8.3) which states

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Theorem (Non-commutative Schur product criterion) A (possibly non-commutative) fusion ring \(R\) is unitarily categorifiable if and only if for all triples of irreducible unital \(*\)-representations \(\left(\pi_s, V_s\right)_{s=1,2,3}\) of \(R\) over \(\mathbb{C}\), and for all \(v_s \in V_s\), we have

\[\sum_i \frac{1}{d\left(x_i\right)} \prod_{s=1}^3\left(v_s^* \pi_s\left(b_i\right) v_s\right) \geq 0\]

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References

Many of these cirteria are listed in Classification of Grothendieck rings of complex fusion categories of multiplicity one up to rank six