Tag - Tilting modules (algebraic groups)

Let G be a simple, simply connected algebraic group defined over a field of positive characteristic p, G_r be its rth Frobenius kernel, and, for q = p^r, G(q) denotes the group of rational points over a field with q elements.

Motivated by work of Curtis and Steinberg, who showed that the simple G_r- and the simple G(q)-modules can be lifted to G, Humphreys and Verma conjectured that the projective covers of the simple G_r-modules also afford a G-module structure. Donkin later refined this conjecture by suggesting that these G_r-projectives lift uniquely to G in the form of tilting modules.

Ballard and Jantzen verified Donkin’s Tilting Module Conjecture for primes that are roughly twice the Coxeter number of the underlying root system or larger. But it was shown by Nakano and his collaborators that the conjecture fails in general. Counterexamples exist for all root systems with the exception of types B₂, where the conjecture holds, and type A, where the conjecture remains completely open.

In this talk we delve into the rich history of these and closely related conjectures and report on their current status.

In this talk, I will start by presenting historical background on the lifting of G-structures on modules for the corresponding Lie algebra. Early work was initiated in the 1960s with fundamental results by Curtis and Steinberg for simple modules. Later questions involving projective modules was studied by Humphreys-Verma, Ballard and Jantzen.

I will then 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 A_n or B₂. Later, I will also present a complete answer to Donkin’s Tilting Module Conjecture for rank 2 groups.

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 A_n or B₂. Later, I will also present a complete answer to Donkin’s Tilting Module Conjecture for rank 2 groups.

The modular representation theory of reductive algebraic groups (general linear groups being an example of such groups) has a number of longstanding open problems. Several of these problems have conjectured resolutions that involve special modules known as indecomposable tilting modules. In this talk we will look at how tilting modules relate to two problems in particular:

   1.  lifting representations from the Lie algebra to the algebraic group, and
   2.  finding a character formula for the irreducible representations.

A good deal of background material will be provided throughout.