University of Sydney Algebra Seminar

Hamilton Wan

March 20 March, 12-1pm, in Carslaw 175

Complex Rank Parabolic Category \(\mathcal{O}\) via Tensor Product Categorifications

Deligne’s categories are symmetric tensor categories that provide a setting for interpolating classical representation-theoretic constructions beyond integer rank. In this talk, I will discuss a complex rank variant of the BGG category O inside Deligne’s categories and explain how, under suitable assumptions, the characters of simple objects can be described in terms of stable limits of parabolic Kazhdan—Lusztig polynomials. In fact, these character formulas are shadows of a deeper structural result: along certain locally closed strata in parameter space, the complex rank categories O are all equivalent to each other, and actually equivalent to a stable limit of classical parabolic categories O. The guiding philosophy is the rigidity of highest weight categories equipped with a compatible categorical action of the Lie algebra \(\mathfrak{sl}_\infty\). I will explain how the complex rank categories O fit into this categorification framework, and time permitting, I will sketch the corresponding uniqueness result for categorifications of tensor products of highest and lowest weight Fock spaces.