The Littelmann path model via the affine Grassmannian
Final Report Abstract
The study of the affine Grassmannian is a very active field of research within geometric representation theory, which possesses intertwining connections with many other areas of mathematics such as number theory, geometric group theory, invariant theory, algebraic combinatorics and algebraic geometry. The problem of obtaining a closed, positive formula for Kostka-Foulkes polynomials is very important in algebraic combinatorics and geometric representation theory. The governing aim in this area of study is to obtain new information about representations of semisimple Lie algebras (also in characteristic p) as well as to connect different constructions which have led researchers to obtain this information (character formulas, branching multiplicities, etc.). The most important finished goal of this project is Theorem 1, which presents a new approach and substantial evidence that it will lead to a full reformulation and proof of Lecouvey’s (still conjectural for arbitrary weight ) formula for symplectic Kostka-Foulkes polynomials in full generality. Our investigations concerning MV polytopes and the affine grassmannian show that the existing seemingly complicated and technical realizations of MV cycles can be described by the simple (literal) unfolding of galleries in an affine building. These findings are a small part of something enormous, and many questions still stand in the horizon.
Publications
- Generalized Lakshmibai-Seshadri galleries Oberwolfach Reports 52/2018, pp. 3107-3109, Mathematisches Forschungsinstitut Oberwolfach
Jacinta Torres
