Graded tensor products, Schur positivity and mock theta functions

本项目的主要目标是研究当前代数的梯度张量积的结构，建立有效的组合公式，用于优秀过滤的梯度多重性，并将这些与mock-theta函数理论联系起来。对当前代数有限维表示范畴的兴趣起源于量子仿射代数及其与所谓的量子Yang-Baxter方程的关系；有限维不可约模提供了这个方程的解。Weyl和Demazure模块——这一类中最大的不可分解模块——的研究将数学分支从KLR代数通过PBW退化连接到对称函数、Kostka多项式和Macdonald多项式的组合。这个项目的一个主要部分是致力于研究Weyl和Demazure模块的一般化版本。我们的目的是利用环面代数双梯度模的新技术确定简单李代数不可约表示的融合积/梯度张量积的结构。在此基础上，利用凸多面体中的晶格点参数化融合积中的最高权向量，并推导出Schur正性猜想。我们进一步的目标是发展优秀过滤的代数和组合方面，找到有效的组合公式，以多维点阵路径为其分级多重性，并将生成序列的专业化与mock-theta函数联系起来。

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.