Grant: Discovery Grants Program - Individual

Groups are present in almost any form of Mathematics, as they are associated to the fundamental notion of symmetry. The building blocks of all groups having finitely many elements are the finite simple groups. One of the most celebrated achievements of Mathematics was the complete classification of all finite simple groups. Except for a few sporadic cases -which where last to be found- and two other families, all other finite simple groups turn out to be groups of Lie type. These are algebraic analogues of important geometric objects called Lie groups. When studying a group, one often looks at various of its concrete images, say by matrices, that is, one studies group ``representations"". These tend to offer different points of views of the same group, and provide information that is sometimes harder to obtain from the group itself. Now all finite groups of Lie type possess a distinguished representation, called Steinberg representation, which reflects in a rich and subtle way the structure of the group. I investigate this Steinberg representation, not only in its original version, using complex numbers, but also in a less understoon version, its so called ``modular reduction"", trying to unlock some its misteries. For instance, in its ordinary version, the Steinberg representation cannot be decomposed into smaller representations, but in its modular version, it is not at all clear how to decompose the Steinberg representation into its irreducible constituents, what these are, or even how many are there. These matters tend to depend not only on delicate properties of the specific group of Lie type in question, but also on the prime used in the reduction.