Representations and binomial coefficients

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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.