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Seminars

Colloquium

On Donkin's Conjectures

2025/06/12 (Thu.) 15:30~16:30
  • Date : 2025/06/12 (Thu.) 15:30~16:30
  • Location : Auditorium, 6 Floor, Institute of Mathematics (NTU Campus)
  • Speaker : Daniel K Nakano (University of Georgia)
  • Abstract :
    I will begin by presenting background material on two famous 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). For over 30 years, these conjectures have eluded verification for primes smaller than $2h-2$ where $h$ is the Coxeter number of the underlying algebraic group.

    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 later describe how we produced infinite families of counterexamples to Jantzen's Question and Donkin's Tilting Module Conjecture. Counterexamples can be produced via our methods for all groups other than when the root system is of type $\rm{A}_{n}$ or $\rm{B}_{2}$. Furthermore, I will also present a complete answer to Donkin's Tilting Module Conjecture for rank 2 groups. At the end of the talk, I will show a few results that indicate that there might be a positive answer to the Tilting Module Conjecture for groups whose underlying root system is of type $A_n$.

    These results represent joint work with Christopher Bendel, Cornelius Pillen and Paul Sobaje.
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