Computing the center of a fusion category

Fabian M\"aurer; Ulrich Thiel

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Computing the center of a fusion category

Fabian Mäurer, Ulrich Thiel

Representation Theory · Submitted 19 Jun 2024

abstract compiled · 2026-10-16 · no language-model review

The authors' main claim We present an algorithm for explicitly computing the categorical (Drinfeld) center of a pivotal fusion category.

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What the abstract itself states, as probabilities: claims first 0.05 · states limitation 0.25 · human participants 0.01 · controlled comparison 0.01 · randomized assignment 0.01 · states quantitative result 0.97 · compares with prior results 0.13 · mentions code or data availability 0.75.

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arXiv listing abstract v3. Results and the main claim are underlined; show every role.

We present an algorithm for explicitly computing the categorical (Drinfeld) center of a pivotal fusion category. Our approach is based on decomposing the images of simple objects under the induction functor from the category to its center. We have implemented this algorithm in a general-purpose software framework TensorCategories for tensor categories that we develop within the open-source computer algebra system OSCAR. We compute explicit models for the centers in form of the tuples $(X,\gamma)$ where $X$ is an object and $\gamma$ is a half-braiding. From these models we can compute the $F$-symbols and $R$-symbols. Using the data from the AnyonWiki, we were able to compute the center together with its $F$-symbols and $R$-symbols for all the 279 multiplicity-free fusion categories up to rank 5, and furthermore some chosen examples of rank 6, including the Haagerup subfactor (presented in a separate paper). Some of these centers yield new high-rank and not multiplicity-free examples of modular categories, e.g. rank 21 and multiplicity 2.

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What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We present an algorithm for explicitly computing the categorical (Drinfeld) center of a pivotal fusion category.	contribution 1.00	0.95	0.57	asserted 1.00
s1	Our approach is based on decomposing the images of simple objects under the induction functor from the category to its center.	method 1.00	0.81	0.06	asserted 1.00
s2	We have implemented this algorithm in a general-purpose software framework TensorCategories for tensor categories that we develop within the open-source computer algebra system OSCAR.	method 0.58	0.95	0.05	asserted 1.00
s3	We compute explicit models for the centers in form of the tuples $(X,\gamma)$ where $X$ is an object and $\gamma$ is a half-braiding.	method 0.74	0.93	0.13	asserted 1.00
s4	From these models we can compute the $F$-symbols and $R$-symbols.	method 0.72	0.86	0.06	asserted 0.78
s5	Using the data from the AnyonWiki, we were able to compute the center together with its $F$-symbols and $R$-symbols for all the 279 multiplicity-free fusion categories up to rank 5, and furthermore some chosen examples of rank 6, including the Haagerup subfactor (presented in a separate paper).	result 0.94	0.95	0.31	asserted 1.00
s6	Some of these centers yield new high-rank and not multiplicity-free examples of modular categories, e.g. rank 21 and multiplicity 2.	result 0.97	0.91	0.18	asserted 0.97

What kind of paper the classifier reports this is: method or tool 1.00 · evidence from simulation or computation 0.98.

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Original abstract

We present an algorithm for explicitly computing the categorical (Drinfeld) center of a pivotal fusion category. Our approach is based on decomposing the images of simple objects under the induction functor from the category to its center. We have implemented this algorithm in a general-purpose software framework TensorCategories for tensor categories that we develop within the open-source computer algebra system OSCAR. We compute explicit models for the centers in form of the tuples $(X,\gamma)$ where $X$ is an object and $\gamma$ is a half-braiding. From these models we can compute the $F$-symbols and $R$-symbols. Using the data from the AnyonWiki, we were able to compute the center together with its $F$-symbols and $R$-symbols for all the 279 multiplicity-free fusion categories up to rank 5, and furthermore some chosen examples of rank 6, including the Haagerup subfactor (presented in a separate paper). Some of these centers yield new high-rank and not multiplicity-free examples of modular categories, e.g. rank 21 and multiplicity 2.

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