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Abstract: Let be a linear differential operator, where is an effective algebraically closed subfield of . It can be shown that the differential Galois group of  is generated (as a closed algebraic group) by a finite number of monodromy matrices, Stokes matrices and matrices in local exponential groups. Moreover, there exist fast algorithms for the approximation of the entries of these matrices. In this paper, we present a numeric–symbolic algorithm for the computation of the closed algebraic subgroup generated by a finite number of invertible matrices. Using the above results, this yields an algorithm for the computation of differential Galois groups, when computing with a sufficient precision. Even though there is no straightforward way to find a “sufficient precision” for guaranteeing the correctness of the end result, it is often possible to check a posteriori whether the end result is correct. In particular, we present a non-heuristic algorithm for the factorization of linear differential operators. [Copyright &y& Elsevier]

Abstract: Let be a finite group. It is easy to compute the character of corresponding to a given complex representation, but much more difficult to compute a representation affording a given character. In part this is due to the fact that a class of equivalent representations contains no natural canonical representation. Although there is a large literature devoted to computing representations, and methods are known for particular classes of groups, we know of no general method which has been proposed which is practical for any but small groups. We shall describe an algorithm for computing an irreducible matrix representation which affords a given character of a given group . The algorithm uses properties of the structure of which can be computed efficiently by a program such as GAP, theoretical results from representation theory, theorems from group theory (including the classification of finite simple groups), and linear algebra. All results in this paper have been implemented in the GAP package REPSN. [Copyright &y& Elsevier]