Representations and binomial coefficients

表示理论研究形式数学系统中对称性的发生。申请人目前主要对正特征的量子群和代数群的表示理论感兴趣。2014年，George Lusztig（麻省理工学院，美国）提出了存在新的表征理论的想法，该理论在量子群（在单位根）的表征和正特征代数群的表征之间进行插值。在过去几年中，至少有三个候选理论被提出：一个由Lusztig提出，另一个由Yang和Zhao（澳大利亚墨尔本）提出，第三个由申请人提出。这三种理论都没有达到预期的效果。该项目的主要目的是结合和统一这三种方法，从而更好地理解新的理论。在这方面最有希望的似乎是研究由正式群体定律产生的新的二项式系数。

Representation theory studies the occurrence of symmetries in formal, mathematical systems. The applicant is currently mostly interested in the representation theories of quantum groups and algebraic groups in positive characteristics. In 2014 George Lusztig (MIT, USA) came up with the idea of the existence of new representation theories that interpolate between representations of quantum groups (at roots of unity) and of algebraic groups in positive characteristics. At least three candidates for these theories were proposed in the last few years: one by Lusztig, another by Yang and Zhao (Melbourne, Australia), and a third by the applicant. Neither of the three theories has yet delivered what is expected. The prinicpal aim of the project is to combine and to unify these three approaches and thus gain a better understanding of the new theories. Most promising for this seems to be the study of new binomial coefficients that arise from formal group laws.