Symmetric groups and geometric representation theory

The project is set in pure mathematics in the areas of representation theory of associative algebras and Lie theory. Representation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces. A representation makes an abstract algebraic object concrete by describing its elements by matrices and the algebraic operations in terms of matrix addition and matrix multiplication. Representation theory is a powerful tool because it reduces problems in abstract algebra to problems in linear algebra, which is well understood. The algebraic objects that can be represented in such a way include groups, associative algebras and Lie algebras. One of the most important classes of associative algebras are group algebras. Their structure depends on the structure of the group involved. If the group is finite, then the group algebra is of finite dimension. This means that, as a vector space, this group algebra has finite dimension. Such an algebra can be written as a product of algebras that cannot be reduced any further. These indecomposable summands are called blocks. The goal of our project is to contribute to the structure theory of the blocks of group algebras of symmetric groups with non-abelian defect groups. Recent developments from Lie theory and higher representation theory opened up completely new perspectives. The inspiration for the current project comes from the connections of the representation theory of symmetric groups to the representation theory of Kac-Moody algebras. Also, the introduction of cluster algebras motivated by phenomena in canonical bases and cluster algebras have led to the discovery of exciting links to many areas of mathematics. Our approach to the above families of algebras involves representation theoretical, combinatorial, homological, geometrical and computational methods. These methods represent a combination of classical methods, whose origin is in the work of James, and new methods originating in Kac-Moody algebras and quantum groups. Justification of such a choice of methods lies in the fact that this is one of the ground-breaking approaches that has the potential to produce very important results in a short time span. This approach is still in its development and it represents one of the most beautiful concepts in modern algebra. Moreover, methods used thus far to study representation theory of symmetric groups did not give desired results. It seems that it is necessary to use higher representation theory and the categorifications approach, as well as geometric realizations to get general results on blocks of symmetric groups and Hecke algebras. This is the advantage of these new and powerful methods over classical methods previously used. This is the right time to use the momentum of development of these concepts in order to get deep results on representation theory of symmetric groups.

The project was conducted in pure mathematics in the areas of representation theory of associative algebras and Lie theory. The aim of the project was to study representations of symmetric groups, cluster algebras and other related associative algebras. The main focus was on the graded structures on these algebras. One of the main results of this project includes a determination of all finite dimensional associative algebras that do not allow non- trivial gradings, [2]. The other major result is the computation of all extensions between rank1 Cohen-Macaulay modules for an algebra categorifying Grassmannian cluster algebras, [1], which gives us first steps in the direction of constructing Auslander-Reiten quiver of this cluster category.Representation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces. A representation makes an abstract algebraic object concrete by describing its elements by matrices and the algebraic operations in terms of matrix addition and matrix multiplication. Representation theory is a powerful tool because it reduces problems in abstract algebra to problems in linear algebra, which is well understood. The algebraic objects that can be represented in such a way include groups, associative algebras and Lie algebras, in particular group algebras. Their structure depends on the structure of the group involved. If the group is finite, then the group algebra is of finite dimension: as a vector space, this group algebra has finite dimension. Such an algebra can be written as a product of algebras that cannot be reduced any further. These indecomposable summands are called blocks. During our project we contributed to the structure theory of the blocks of algebras.Recent developments from Lie theory and higher representation theory opened up completely new perspectives. The inspiration for our project came from the connections of representation theory of symmetric groups to the representation theory of Kac-Moody algebras and between canonical bases and cluster algebras. Our approach to the above families of algebras involved representation theoretical, combinatorial, homological and geometrical methods. These methods represent a combination of classical and new methods, with the potential to produce important results in a short time span.References:[1] Extension between CM-modules for Grassmannian cluster categories, to appear in J.Alg. Comb, arxiv:1601.05943.[2] Existence of gradings on associative algebras, Volume 44, 2016 - Issue 7, (2016), 3069-3076. Green OA