Representation Theory of Reflection Groups
Dr. Hery Randriamaro (Universität Kassel)
It all started in 1896, when Dedekind sent a letter to Frobenius. He wrote about the group determinant, explained how it factors in the abelian case, and suggested him to think about the nonabelian case. It is the question of factoring the group determinant of an arbitrary group that gave rise to representation theory by Frobenius. This latter initially developed the representation of groups essentially based on the character groups. But over time, the theory has provided groups with concrete descriptions in terms of linear algebra. Namely, each element of a group is represented by a matrix in such a way that the group operation is matrix multiplication. The most studied representation until now is that of reflection groups. A reflection is a map on a structured object preserving its structure. These groups arise in a multitude of ways in mathematics. Moreover, many groups are isomorphic to some reflection groups. Classical examples are the Coxeter groups, the symmetry groups of regular polytopes, and the symmetric groups. These latter are particularly important as the Cayley theorem states that every group is isomorphic to a subgroup of a symmetric group. And reflection groups may be regarded as the foundation of other algebraic structures. Like the descent algebras which are subalgebras of the group algebras of reflection groups, and the Hecke algebras which are deformations of the group algebras of reflection groups. Representation theory naturally extends to representation theory of algebras since every group can be extended to group algebras. The presentation is essentially divided into three parts: the origin of representation theory, representation theory of reflection groups and related algebras, and applications of representations of groups and related algebras. Besides, it is good to warn the experts on the subject that the presentation is aimed to be instructive, in the sense that enough time is planned to be spent on basic definitions.