Finite groups appear naturally as sets of symmetries. As such, their study, which started in the early 19th century, is of fundamental importance, and they have applications in many different domains, such as biology, physics or chemistry. They also appear in various areas across mathematics.
While groups can be studied abstractly, it is often easier or more interesting to consider them through the way they actually act, especially on particular objects such as vector spaces. Vector spaces are very familiar frameworks, like the 3-dimensional space around us, Einstein's 4-dimensional space-time, or the 11-dimensional space needed for M-theory in string theory.
The study of group actions on vector spaces is known as representation theory, and it has been an active research area in mathematics since the beginning of the 20th century, with ramifications throughout mathematics and other sciences.
Of particular interest is the symmetric group, which contains all the possible permutations of a given set of objects. It has long been known that the representation theory of the symmetric group can be described using very elegant combinatorial tools, like integer partitions (the different ways to write a given positive integer as a sum of smaller integers).
The study of representations becomes, however, much harder when we let the groups act on vector spaces over fields of prime characteristic, as opposed to the more natural characteristic 0 of the fields of rational, real or complex numbers. In this context, we talk about modular representations, while ordinary representations relate to fields of characteristic 0. One long-standing problem in representation theory is the determination of the decomposition matrix, which enlightens the relationship between ordinary and modular representations.
Much work has been done in recent years to try and understand the modular representations of the symmetric groups, and of other related groups (such as the alternating groups, or Coxeter groups and complex reflection groups). This often involves the study of other objects, like Iwahori-Hecke algebras or quantum groups, whose representations can then be connected to those of the symmetric group.
The approach we take in this project is to use as much as possible of the well-understood ordinary representation theory to produce modular information. One of our main tools is given by so-called basic sets. These can be found in characteristic 0 and used to provide insight into the case of prime characteristic. We also make heavy use of powerful correspondences between groups, called perfect isometries. These can be used to transport much of the representation theory, in particular basic sets, thereby allowing us to translate to complicated settings results we can readily obtain in smaller or more easily handled groups.
The first part of the project will be devoted to exhibiting a large collection of perfect isometries, most notably between families of complex reflection groups, and some of their natural subgroups. These perfect isometries will then be used to describe basic sets in various groups related to the symmetric group. Finally, these basic sets will be studied and compared, in order to derive as much information as possible about the modular representation theory of these groups.
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