In this talk, I will present background material on two conjectures formulated by Donkin at MSRI in 1990. The first conjecture is Donkin’s Tilting Module Conjecture (DTilt), and the second conjecture is Donkin’s p-Filtration Conjecture (DFilt). Recent progress by Kildetoft-Nakano and Sobaje has shown that there are important connections between these conjectures. In particular, Jantzen’s Question posed in 1980 on the existence of Weyl p-filtrations for Weyl modules for a reductive algebraic group constitutes a central part of the new developments. I will describe how we produced infinite families of counterexamples to Jantzen’s Question and Donkin’s Tilting Module Conjecture. New techniques to exhibit explicit examples are provided along with methods to produce counterexamples in large rank from counterexamples in small rank. Counterexamples can be produced via our methods for all groups other than when the root system is of type An or B2. Later, I will also present a complete answer to Donkin’s Tilting Module Conjecture for rank 2 groups.
These results represent joint work with Christopher Bendel, Cornelius Pillen and Paul Sobaje.
This video is part of the University of Georgia‘s Algebra seminar.
