Graded tensor products, Schur positivity and mock theta functions

这个项目的主要目标是研究当前代数的分次张量积的结构，建立有效的组合公式的分级多重性在优秀的过滤和连接这些理论的mock-theta函数。感兴趣的范畴有限维表示的电流代数起源于背景下的量子仿射代数和它们的关系，以所谓的量子杨-巴克斯特方程;有限维不可约模提供该方程的解。Weyl和Demazure模是这一范畴中最大的不可分解模，它的研究将数学分支从KLR代数通过PBW退化连接到对称函数的组合学、Kostka多项式和Macdonald多项式。本项目的一个主要部分致力于Weyl和Demazure模的广义版本的研究。我们的目标是确定结构的融合产品/分次张量积的不可约表示的简单李代数使用新的技术从双分次模的环面代数。此外，我们的目标是参数化的最高权重向量的融合产品的凸多面体的格点，并推导出Schur正猜想。我们的进一步目标是发展的代数和组合方面的优秀filtrations，找到有效的组合公式，其分级的多重性方面的多维格路径和连接专业化的生成系列模拟θ函数。

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.