Graded tensor products, Schur positivity and mock theta functions

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The main goals of this project are to investigate the structure of graded tensor products for current algebras, to establish effective combinatorial formulas for graded multiplicities in excellent filtrations and to connect these with the theory of mock-theta functions.The interest in the category of finite dimensional representations of current algebras originated in the context of quantum affine algebras and their relation to the so-called quantum Yang-Baxter equation; finite–dimensional irreducible modules provide solutions to this equation. The study of Weyl and Demazure modules - the largest indecomposable modules in this category - connects branches of mathematics from KLR algebras via PBW degenerations to combinatorics of symmetric functions, Kostka polynomials and Macdonald polynomials.One major part of this project is devoted to the study of generalized versions of Weyl and Demazure modules. We aim to determine the structure of fusion products/graded tensor products of irreducible representations of simple Lie algebras using new techniques from bi–graded modules for toroidal algebras. Furthermore, we aim to parametrize the highest weight vectors in fusion products by lattice points in convex polytopes and to derive the Schur positivity conjecture. Our further objectives are to develop the algebraic and combinatorial aspects of excellent filtrations, to find effective combinatorial formulas for their graded multiplicities in terms of multidimensional lattice paths and connect specializations of generating series to mock-theta functions.