Are there better upper bounds on the rank of the commutant of a fusion module than the global dimension? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-26T04:35:25Z http://mathoverflow.net/feeds/question/105208 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/105208/are-there-better-upper-bounds-on-the-rank-of-the-commutant-of-a-fusion-module-tha Are there better upper bounds on the rank of the commutant of a fusion module than the global dimension? Scott Morrison 2012-08-22T04:45:22Z 2012-08-23T04:01:03Z <p>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 $\mathcal{C}$.</p> <blockquote> <p>Can I bound the rank of $\mathcal{C}_\mathcal{M}^*$?</p> </blockquote> <p>Recall that the <i>rank</i> 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?</p> <p><em>Update</em>: 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 $\mathcal{C}$.</p> <p>As examples:</p> <ul> <li>the Haagerup subfactor gives a Morita equivalence between two fusion categories with ranks 4 and 6, and global dimension $\approx 35.725$</li> <li>$\operatorname{Rep}(G)$ 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.</li> </ul> http://mathoverflow.net/questions/105208/are-there-better-upper-bounds-on-the-rank-of-the-commutant-of-a-fusion-module-tha/105243#105243 Answer by Noah Snyder for Are there better upper bounds on the rank of the commutant of a fusion module than the global dimension? Noah Snyder 2012-08-22T14:50:15Z 2012-08-22T14:50:15Z <p>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 <a href="http://arxiv.org/abs/0812.2590" rel="nofollow">"Finite groups have even more conjugacy classes"</a>). I learned about this from Pavel Etingof when Eric Rowell asked him about a similar question for ranks of centers.</p> <p>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?</p> http://mathoverflow.net/questions/105208/are-there-better-upper-bounds-on-the-rank-of-the-commutant-of-a-fusion-module-tha/105277#105277 Answer by Evan Jenkins for Are there better upper bounds on the rank of the commutant of a fusion module than the global dimension? Evan Jenkins 2012-08-22T23:38:46Z 2012-08-23T04:01:03Z <p>If $\mathcal{C}$ and $\mathcal{D}$ are Morita equivalent by a pair <code>$({_\mathcal{C}}\mathcal{M}_{\mathcal{D}}, {_\mathcal{D}}\mathcal{N}_{\mathcal{C}})$</code> of bimodules, then there is a natural map of fusion bimodules <code>${_D}N \otimes_{C} M_{D} \to {_D}D_D$</code> that preserves Frobenius-Perron dimension (see section 5.1 of <a href="http://front.math.ucdavis.edu/1202.4396" rel="nofollow">Noah's latest preprint with Pinhas Grossman</a>). 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 <a href="http://front.math.ucdavis.edu/1004.0665" rel="nofollow">your paper with Noah and Frank Calegari</a>) to determine some nontrivial lower bounds on Frobenius-Perron dimensions of simple objects in $\mathcal{D}$, and thus improve on the global dimension bound.</p> <p>(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.)</p>