2025/01/27 by Alexandra Utiralova, Utiralova, Alexandra
Mathematics · #Finite Group Theory Research
paper · pdf · doi:10.48550/arxiv.2501.15778
Following the work of Venkatesh (arXiv:2203.03158), we study further the categories of representations of the general linear groups GL(X) in the Verlinde category Verp in characteristic p. The main question we answer is how to translate between highest weight labelings for different choices of the Borel subgroup B(X)⊂ GL(X). We do this by reducing the general case to the study of representations of the group GL(X) for X=Lm⊕ Ln using the method of odd reflections. On the category of representations of GL(Lm⊕ Ln) we introduce the structure of the highest weight category, as well as the categorical action of \widehat\mathfrakslp through translation functors. It allows us to understand projective and injective objects, BGG reciprocity, duality and lowest weights for simple modules, and standard filtration multiplicities for projective objects.