A Note on Applications of the ‘vector Enumerator’ Algorithm

It is one of the aims of computational representation theory to provide tools for the explicit construction of representations of abstractly given algebras. If the algebra under consideration is given by a finite algebra presentation and the representation searched for is given by a finite module presentation, then the algorithmic tool at hand is the so-called VectorEnumerator algorithm, see Section 1.1 and Theorem 1.2. In this note we are going to show how it is possible to exploit certain abstract situations to obtain algebra or module presentations, making them accessible for analysis using the VectorEnumerator algorithm. In Section 2 we show how module presentations can be obtained in two particular situations. Firstly an explicit matrix representation can be used to give a module presentation, and secondly we consider so-called local modules. In Section 3 we show how presentations of induced modules and of certain epimorphic images are found. In Section 4 we are concerned with field extensions and how these affect algebra presentations. In the final Section 5 we show by example, a so-called Iwahori-Hecke algebra over a certain cyclotomic field, how the techniques described in this note work together in practice. Iwahori-Hecke algebras have gained considerable interest in the modular representation theory of finite groups of Lie type. In particular for the exceptional types methods of computational representation theory have been of great help in understanding the representation theory of Iwahori-Hecke algebras. In fact, examples of this kind have been the original motivation for the present work. The example presented here is part of a broader examination of exceptional IwahoriHecke algebras done by the author in [11]. Recently, there is rising interest in certain generalizations of Iwahori-Hecke algebras, the so-called cyclotomic Hecke algebras, see e.g. [2]. Again there are exceptional types, which for the time being defy deeper theoretical analysis. The computational techniques described in this note again turn out to be very helpful to understand the structure and the representation theory of yclotomic Hecke algebras. At the moment this is work under progress and details on this will appear elsewhere. Let us now begin by setting the stage for the main actor, where as a general reference see [1, Section III.2.8.].