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