arXiv subjects
Publications and source records attributed to Marc Keilberg.
We study the categorical notion of braid gauging and obtain its classical Hopf algebraic description. We demonstrate how braid gauging can provide new insights on certain categorical invariants, such as the fusion rules and the higher Frobenius-Schur indicators. The running example for the paper is the category $\operatorname{Rep}(D(G))\cong \mathcal{Z}(\text{Vec}_G)$, whose braid gaugings are studied in-depth.
Let $p$ be an odd prime with $p\equiv1\bmod 4$. Then for any odd power $q$ of $p$ and a positive integer $j$ we show that the groups $\text{Sp}_{p^j+1}(q),\text{PSp}_{p^j+1}(q)$, and their Sylow $p$-subgroups are non-$FSZ_{p^j}$.
We give a classification of all quasitriangular structures and ribbon elements of $\mathcal{D}(G)$ explicitly in terms of group homomorphisms and central subgroups. This can equivalently be interpreted as an explicit description of all braidings with which the tensor category $\operatorname{Rep}(\mathcal{D}(G))$ can be endowed. We also characterize their equivalence classes under the action of $\operatorname{Aut}(\mathcal{D}(G))$ and determine when they are factorizable.
We show that any group $G$ with a non-$FSZ_m$ quotient by a central cyclic subgroup also provides a non-$FSZ_m$ group of order $m|G|$ obtained as a central product of $G$ with a cyclic group. We then construct, for every prime $p>3$ and $j\in\mathbb{N}$, an $FSZ_{p^j}$ group $F$ such that there is a central cyclic subgroup $A$ with $F/A$ not $FSZ_{p^j}$. We apply these results to regular wreath products to construct an $FSZ$ $p$-group which is not $FSZ^+$ for any prime $p>3$. These give the first known examples of $FSZ$ groups that are not $FSZ^+$. We are also able to prove a few partial results concerning the $FSZ$ properties for the Sylow subgroups of symmetric groups. In the appendix we enumerate all non-$FSZ$ groups of order $5^7$.
For any prime $p>3$ and $j\in\mathbb{N}$ we construct examples of non-$FSZ_{p^j}$ groups of order $p^{p^j+2j-1}$. In the special case of $j=1$ this yields groups of order $p^{p+1}$, which is the minimum possible order for a non-$FSZ$ $p$-group.
We investigate a possible connection between the $FSZ$ properties of a group and its Sylow subgroups. We show that the simple groups $G_2(5)$ and $S_6(5)$, as well as all sporadic simple groups with order divisible by $5^6$ are not $FSZ$, and that neither are their Sylow 5-subgroups. The groups $G_2(5)$ and $HN$ were previously established as non-$FSZ$ by Peter Schauenburg; we present alternative proofs. All other sporadic simple groups and their Sylow subgroups are shown to be $FSZ$. We conclude by considering all perfect groups available through GAP with order at most $10^6$, and show they are non-$FSZ$ if and only if their Sylow 5-subgroups are non-$FSZ$.
We prove several results concerning quasi-bialgebra morphisms $\mathcal{D}^ω(G)\to\mathcal{D}^η(H)$ of twisted group doubles. We take a particular focus on the isomorphisms which are simultaneously isomorphisms $\mathcal{D}(G)\to\mathcal{D}(H)$. All such isomorphisms are shown to be morphisms of quasi-Hopf algebras, and a classification of all such isomorphisms is determined. Whenever $ω\in Z^3(G/Z(G),U(1))$ this suffices to completely describe $\operatorname{Aut}(\mathcal{D}^ω(G))$, the group of quasi-Hopf algebra isomorphisms of $\mathcal{D}^ω(G)$, and so generalizes existing descriptions for the case where $ω$ is trivial.
We prove a variety results on tensor product factorizations of finite dimensional Hopf algebras (more generally Hopf algebras satisfying chain conditions in suitable braided categories). The results are analogs of well-known results on direct product factorizations of finite groups (or groups with chain conditions) such as Fitting's Lemma and the uniqueness of the Krull-Remak-Schmidt factorization. We analyze the notion of normal (and conormal) Hopf algebra endomorphisms, and the structure of endomorphisms and automorphisms of tensor products. The results are then applied to compute the automorphism group of the Drinfeld double of a finite group in the case where the group contains an abelian factor. (If it doesn't, the group can be calculated by results of the first author.)
Using a recent classification of $\operatorname{End}(\mathcal{D}(G))$, we determine a number of properties for $\operatorname{Aut}(\mathcal{D}(G))$, where $\mathcal{D}(G)$ is the Drinfel'd double of a finite group $G$. Furthermore, we completely describe $\operatorname{Aut}(\mathcal{D}(G))$ for all purely non-abelian finite groups $G$. A description of the action of $\operatorname{Aut}(\mathcal{D}(G))$ on $\operatorname{Rep}(\mathcal{D}(G))$ is also given. We are also able to produce a simple proof that $\mathcal{D}(G)\cong\mathcal{D}(H)$ if and only if $G\cong H$, for $G$ and $H$ finite groups.
We investigate the indicators for certain groups of the form $\BZ_k\rtimes D_l$ and their doubles, where $D_l$ is the dihedral group of order $2l$. We subsequently obtain an infinite family of totally orthogonal, completely real groups which are generated by involutions, and whose doubles admit modules with second indicator of -1. This provides us with answers to several questions concerning the doubles of totally orthogonal finite groups.
In this paper we explicitly determine all indicators for groups isomorphic to the semidirect product of two cyclic groups by an automorphism of prime order, as well as the generalized quaternion groups. We then compute the indicators for the Drinfel'd doubles of these groups. This first family of groups includes the dihedral groups, the non-abelian groups of order $pq$, and the semidihedral groups. We find that the indicators are all integers, with negative integers being possible in the first family only under certain specific conditions.