Generalised sifting in black-box groups

We present a generalisation of the sifting procedure introduced originally by Sims for computation with finite permutation groups, and now used for many computational procedures for groups, such as membership testing and finding group orders. Our procedure is a Monte Carlo algorithm, and is presented and analysed in the context of black-box groups. It is based on a chain of subsets instead of a subgroup chain. Two general versions of the procedure are worked out in detail, and applications are given for membership tests for several of the sporadic simple groups. Our major objective was that the procedures could be proved to be Monte Carlo algorithms, and their costs computed. In addition we explicitly determined suitable subset chains for six of the sporadic groups, and we implemented the algorithms involving these chains in the {\sf GAP} computational algebra system. It turns out that sample implementations perform well in practice. The implementations will be made available publicly in the form of a {\sf GAP} package

Formulas for primitive Idempotents in Frobenius Algebras and an Application to Decomposition maps

In the first part of this paper we present explicit formulas for primitive idempotents in arbitrary Frobenius algebras using the entries of representing matrices coming from projective indecomposable modules with respect to a certain choice of basis. The proofs use a generalisation of the well known Frobenius-Schur relations for semisimple algebras. The second part of this paper considers \Oh-free \Oh-algebras of finite \Oh-rank over a discrete valuation ring \Oh and their decomposition maps under modular reduction modulo the maximal ideal of \Oh, thereby studying the modular representation theory of such algebras. Using the formulas from the first part we derive general criteria for such a decomposition map to be an isomorphism that preserves the classes of simple modules involving explicitly known matrix representations on projective indecomposable modules. Finally we show how this approach could eventually be used to attack a conjecture by Gordon James in the formulation of Meinolf Geck for Iwahori-Hecke-Algebras, provided the necessary matrix representations on projective indecomposable modules could be constructed explicitly.Comment: 16 page

Polynomial-time proofs that groups are hyperbolic

Funding: UK EPSRC grant number EP/I03582X/1.It is undecidable in general whether a given finitely presented group is word hyperbolic. We use the concept of pregroups, introduced by Stallings (1971), to define a new class of van Kampen diagrams, which represent groups as quotients of virtually free groups. We then present a polynomial-time procedure that analyses these diagrams, and either returns an explicit linear Dehn function for the presentation, or returns fail, together with its reasons for failure. Furthermore, if our procedure succeeds we are often able to produce in polynomial time a word problem solver for the presentation that runs in linear time. Our algorithms have been implemented, and when successful they are many orders of magnitude faster than KBMAG, the only comparable publicly available software.PostprintPeer reviewe

The Crowdsourced Replication Initiative: Investigating Immigration and Social Policy Preferences. Executive Report.

In an era of mass migration, social scientists, populist parties and social movements raise concerns over the future of immigration-destination societies. What impacts does this have on policy and social solidarity? Comparative cross-national research, relying mostly on secondary data, has findings in different directions. There is a threat of selective model reporting and lack of replicability. The heterogeneity of countries obscures attempts to clearly define data-generating models. P-hacking and HARKing lurk among standard research practices in this area.This project employs crowdsourcing to address these issues. It draws on replication, deliberation, meta-analysis and harnessing the power of many minds at once. The Crowdsourced Replication Initiative carries two main goals, (a) to better investigate the linkage between immigration and social policy preferences across countries, and (b) to develop crowdsourcing as a social science method. The Executive Report provides short reviews of the area of social policy preferences and immigration, and the methods and impetus behind crowdsourcing plus a description of the entire project. Three main areas of findings will appear in three papers, that are registered as PAPs or in process