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Suppose I have a fusion category $\mathcal{C}$ and an indecomposable module category $\mathcal{M}$ over it. The commutant $\mathcal{C}_\mathcal{M}^*$ is the category of module endofunctors, and gives another fusion category, Morita equivalent to the original $\mathcal{C}$.

Can I bound the rank of $\mathcal{C}_\mathcal{M}^*$?

Recall that the rank is the number of isomorphism classes of irreducible objects. Certainly $\mathcal{C}$ and $\mathcal{C}_\mathcal{M}^*$ have the same global dimension, so easily $\operatorname{rank}(\mathcal{C}_\mathcal{M}^*) \leq \operatorname{dim}(\mathcal{C})$. Are there better upper bounds available?

Update: I'm happy to consider all the 'decategorified' data of $\mathcal{C}$ and $\mathcal{M}$, that is, the Grothendieck groups of both, along with the ring and module structures thereon, when trying to come up with an estimate, not just the rank of $\mathcal{C}$.

As examples:

  • the Haagerup subfactor gives a Morita equivalence between two fusion categories with ranks 4 and 6, and global dimension $\approx 35.725$
  • $\operatorname{Rep}(G)$ and $\text{Vec}_G$ are Morita equivalent, with global dimension $|G|$. Here $\operatorname{rank}(\text{Vec}_G) = |G|$, while when $G$ is non-commutative $\operatorname{rank}(\operatorname{Rep}(G))$ may be much smaller.
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If $\mathcal{C}$ and $\mathcal{D}$ are Morita equivalent by a pair $({_\mathcal{C}}\mathcal{M}_{\mathcal{D}}, {_\mathcal{D}}\mathcal{N}_{\mathcal{C}})$ of bimodules, then there is a natural map of fusion bimodules ${_D}N \otimes_{C} M_{D} \to {_D}D_D$ that preserves Frobenius-Perron dimension (see section 5.1 of Noah's latest preprint with Pinhas Grossman). So the Frobenius-Perron dimensions of elements of $N \otimes_C M$ (which do not depend on knowing $D$) will give Frobenius-Perron dimensions of elements of $D$. Then I think you may be able to use the known possible small Frobenius-Perron dimensions of objects (from your paper with Noah and Frank Calegari) to determine some nontrivial lower bounds on Frobenius-Perron dimensions of simple objects in $\mathcal{D}$, and thus improve on the global dimension bound.

(Or do arbitrarily small numbers of the form $2 \cos (\pi / n)$ already generate the full ring of real cyclotomic integers? Even if so, this method could at least reduce the bound by 1 for weakly integral categories, although that's not a great improvement.)

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Thanks Evans, this does seem promising at least for some small cases. For example for Haagerup the module has an object of dimension $\alpha = \sqrt{(5+\sqrt{13})/2}$, so $D$ has a (not-necessarily simple) object of dimension $\alpha^2 = (5+\sqrt{13})/2 \simeq 4.30278$. It's easy to see that this can only be decomposed as $1+1+(1+\sqrt{13})/2$, $2+(1+\sqrt{13})/2$ or $2 \cos(\pi/n) + x$, where $x > 76/33$. There's a finite list of such $n$, and it turns out $x$ is never maximal amongst its Galois conjugates except when $n = 2$ or $3$. – Scott Morrison Aug 24 '12 at 2:31
Thus we have to split into four cases already, but in each we've already identified the dimensions of some simple objects in $D$, and hence improved the bound on the rank. – Scott Morrison Aug 24 '12 at 2:32

In the special case of Vec(G) and Rep(G) there's a lot of results in the literature. Usually the phrasing is in terms of conjugacy classes (e.g. the strongest proved bound as far as I know is Keller's "Finite groups have even more conjugacy classes"). I learned about this from Pavel Etingof when Eric Rowell asked him about a similar question for ranks of centers.

Do I understand correctly though that you're happy to have conditions that involve more than just the rank of C? E.g. if I say wanted to have the full list of dimensions of objects in C as input to the bound would that be a problem?

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Thanks Noah for the link. I clarified that I'm happy to use all the information in the fusion ring and its module. – Scott Morrison Aug 22 '12 at 22:01

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