Geometric Representation Theory

The aim of the School is to train young students in research activity in several areas of Geometric Representation Theory. The proposed topics include areas which are currently very active and the School intends to initiate the students in research by proposing them Research Projects under the supervision of senior researchers involved in the School. The school is going to give sufficient background to provide the students with the necessary tools to start their PhD work. The presence of a large number of mathematicians working in Geometric Representation Theory is going to strengthen the impact of this research discipline in Indonesia. The students coming to the School will come from many parts of the greater region and we intend to use the contacts which are certainly going to be established to build on further networks centering around Geometric Representation Theory in South-East Asia. Further we will use the leading role the mathematical faculty of ITB is already playing in Indonesia to advertise the subject in Indonesia more than it is already.

This school is co-sponsored by the Abdus Salam International Centre for Theoretical Physics, Trieste, as an External Activity. Young researchers from developing countries who are members of UNESCO are encouraged to apply.

Geometric representation theory has many different aspects. We will focus on mainly two of them. First, given an algebraic variety with a group acting by homomorphism of varieties on it gives the possibility to use representation theoretic techniques to study the variety as well as the group by means of the action. A particular case appears in the following context. Given a finite dimensional algebra over an algebraically closed field and an integer d, then the representations of the algebra of dimension d are given by matrices satisfying polynomial relations, whence giving a variety. Isomorphism classes are provided by orbits under the conjugation action of the general linear group, and hence the setting is given. The case of the algebra of upper triangular matrices gives flag varieties, a quotient of which is the algebra of complexes of vector spaces. Grassmann varieties appear in this context. A module is said to degenerate to a second module if the second module is in the closure of the adherence of the orbit of the first. An algebraic characterization of this situation has been given. All these aspects are going to be studied in the school and various links between the subjects are going to be explored.