Representations of cyclotomic Hecke algebras

Take a classical and much-loved mathematical object, one which has been studied extensively since the 19th century. We want to look at some particular aspects of its behaviour. We find there are two cases. In one environment, it is well understood; however out of that environment, it is, as yet, largely unpredictable. It obeys a framework of rules, but beyond the restrictions imposed by those rules we have limited knowledge. Our aim is to give as precise a description as possible of the ways in which it can behave. But our classical friend is just one specific case in a class of structures that arise in various contexts in mathematics and physics. Everything in this class exhibits similar properties, although most members have more complicated behaviour than the original object. It may therefore be useful to us to study the entire class, and specialize our results as appropriate. Moreover, it turns out that there are deep connections between these new objects and other, seemingly unrelated, branches of mathematics.

The classical object is the symmetric group and we want to know about its representations. Each representation is like a Lego structure: it is made up of a set of building blocks. The building blocks are the irreducible representations. Over the field of complex numbers, these irreducible representations are understood and classified, and we know precisely how to join them together to make new representations. Thus over the complex numbers, we understand all representations of the symmetric groups. But over an arbitrary field, this is no longer true. We would, however, gain substantial information if we knew about the structure of certain representations, called Specht modules.

The more general objects are the cyclotomic Hecke algebras of type G(r,1,n), of which the symmetric group algebra is a particular case. When the algebra is not semisimple, very little is known about its representations, although they appear in various representation theoretic contexts, and have been shown to have deep links with algebraic Lie theory. The principal aim of this research is to investigate the properties of certain representations of the cyclotomic Hecke algebras. We would like to build on our knowledge of the symmetric group algebras to find information about these representations, but we also want to develop more general results that will take us in new directions.