Higher Indicators for the Doubles of some Totally Orthogonal 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.Comment: 28 page

Regulation of motility and polarity in Myxococcus xanthus

M. xanthus cells possess two independent motility systems: the adventurous (A) system and the social (S) system. S-motility depends on the extension and retraction of Type-4-pili, whereas A-motility is mediated via focal adhesion complexes that incorporate a MotAB-like motor. The rod-shaped M. xanthus cells can reverse the direction of movement, which is accompanied by a polarity inversion of components of both motility systems. Reversals are induced by the Frz chemosensory system, acting upstream of a small GTPase, MglA and its cognate GTPase activating protein, MglB. MglA and MglB localize to opposite cell poles in a moving cell, defining the leading pole (MglA) and the lagging pole (MglB). MglA and MglB directly interact. In this study we identified residues in MglB that are required for the interaction with MglA. Furthermore, we show that inhibition of the MglA/MglB interaction affects MglA GTPase activity and localization of MglB. In addition to the MglA/MglB system, the response regulator RomR is required for motility and reversals. RomR localizes in a bipolar asymmetric pattern with a large cluster at the lagging cell pole. Previously RomR was reported to regulate the A-motility system. We show that RomR localization does not depend on A-motility proteins. In contrast, we found that RomR is required for both motility systems, suggesting that it acts upstream of the two motility machineries. Consistent with that, we found that RomR directly interacts with MglA and MglB. Moreover, RomR, MglA and MglB affect the localization of each other in all pair-wise directions suggesting that RomR stimulates motility by promoting correct localization of MglA and MglB in MglA/RomR and MglB/RomR complexes at opposite poles. Furthermore, localization analyses suggest that the two RomR complexes mutually exclude each other from their respective poles. We further showed that RomR interfaces with FrzZ, the output response regulator of the Frz chemosensory system, to regulate reversals. Thus, RomR serves at the interface to connect a classic bacterial signalling module (Frz) to a classic eukaryotic polarity module (MglA/MglB). This modular design is paralleled by the phylogenetic distribution of the proteins suggesting an evolutionary scheme in which RomR was incorporated into the MglA/MglB module to regulate cell polarity followed by the addition of the Frz system to dynamically regulate cell polarity. Importantly, RomR possesses a conserved aspartate in its receiver domain, required for activation via phosphorylation. Because we found no evidence for direct phosphotransfer between FrzE and RomR, further phylogenetic studies were carried out. These analyzis revealed two candidate proteins involved in motility, RomX and RomY, which display a co-evolutionary relationship with RomR. We show that both proteins are involved in motility and that RomX behaves similarly to RomR with respect to phenotype and localization. We suggest that RomX and RomY play a role in regulation of motility together with RomR, MglA and MglB and possibly in RomR activation

Quasitriangular structures of the double of a finite group

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.Comment: 30 pages; v2: Partial minimality results replaced with complete results on factorizability. Additional details added to some results. Some potential applications, to be considered in a future paper, are also discusse