Scott H. Murray

Computing in groups of Lie type

We describe two methods for computing with the elements of untwisted groups of Lie type: using the Steinberg presentation and using highest weight representations. We give algorithms for element arithmetic within the Steinberg presentation. Conversion between this presentation and linear representations is achieved using a new generalisation of row and column reduction.

Certifying Solutions to Permutation Group Problems

We describe the integration of permutation group algorithms with proof planning. We consider eight basic questions arising in computational permutation group theory, for which our code provides both answers and a set of certificates enabling a user, or an intelligent software system, to provide a full proof of correctness of the answer. To guarantee correctness we use proof planning techniques, which construct proofs in a human-oriented reasoning style. This gives the human mathematician the necessary insight into the computed solution, as well as making it feasible to check the solution for relatively large groups.

Computing in unipotent groups and reductive algebraic groups

The unipotent groups are an important class of algebraic groups. We show that techniques used to compute with finitely generated nilpotent groups carry over to unipotent groups. We concentrate particularly on maximal unipotent subgroups of split reductive groups and show how this improves computation in the reductive group itself.

Selecting base points for the Schreier-Sims algorithm for matrix groups

Abstract We consider the application of the Schreier-Sims algorithm and its variations to matrix groups defined over finite fields. We propose a new algorithm for the selection of the base points and demonstrate that it both improves the performance of the algorithm for a large range of examples and significantly extends the range of application. In particular, the random Schreier-Sims algorithm, with this enhancement, performs extremely well for almost simple groups.

Product Replacement in the Monster

We show that the product replacement algorithm can be used to produce random elements of the Monster group. These random elements are shown to have the same distribution of element orders as uniformly distributed random elements after a small number of steps.

Automatic proof of graph nonisomorphism

Abstract.We describe automated methods for constructing nonisomorphism proofs for pairs of graphs. The proofs can be human-readable or machine-readable. We have developed an experimental implementation of an interactive webpage producing a proof of (non)isomorphism when given two graphs.

Representations of Parabolic and Borel Subgroups

We demonstrate a relationships between the representation theory of Borel subgroups and parabolic subgroups of general linear groups. In particular, we show that the representations of Borel subgroups could be computed from representations of certain maximal parabolic subgroups.