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Categorical Lagrangian Grassmannians and Brauer-Picard groups of pointed fusion categories

Dmitri Nikshych, Brianna Riepel

Abstract

We analyze the action of the Brauer-Picard group of a pointed fusion category on the set of Lagrangian subcategories of its center. Using this action we compute the Brauer-Picard groups of pointed fusion categories associated to several classical finite groups. As an application, we construct new examples of weakly group-theoretical fusion categories.

Categorical Lagrangian Grassmannians and Brauer-Picard groups of pointed fusion categories

Abstract

We analyze the action of the Brauer-Picard group of a pointed fusion category on the set of Lagrangian subcategories of its center. Using this action we compute the Brauer-Picard groups of pointed fusion categories associated to several classical finite groups. As an application, we construct new examples of weakly group-theoretical fusion categories.

Paper Structure

This paper contains 16 sections, 29 theorems, 87 equations.

Table of Contents

  1. Introduction
  2. Conventions and notation
  3. Some facts about cohomology of finite groups
  4. Fusion categories and their Brauer-Picard groups
  5. Parameterization of $\mathop{\mathrm{\operatorname{\mathsf{BrPic}}}}\nolimits(\operatorname{\operatorname{\mathsf{Vec}}}_G)$
  6. Braided autoequivalences of centers
  7. Action on the categorical Lagrangian Grassmannian
  8. Examples: symmetric and alternating groups
  9. Symmetric group $S_3$
  10. Symmetric group $S_4$
  11. Alternating group $A_4$
  12. Examples: non-abelian groups of order $8$
  13. Dihedral group $D_8$
  14. Quaternion group $Q_8$
  15. Examples: groups of order $pq$
  16. ...and 1 more sections

Key Result

Proposition 3.2

Let $G$ be a finite group and let $A$ be a finite $G$-module such that the orders $|G|$ and $|A|$ are relatively prime. Then $H^n(G,\,A)=0$ for all $n$.

Theorems & Definitions (70)

  • Remark 3.1
  • Proposition 3.2
  • Theorem 3.3
  • Theorem 3.4
  • Proposition 4.1
  • Remark 4.2
  • Lemma 5.1
  • proof
  • Proposition 5.2
  • proof
  • ...and 60 more