On good (p,r)-filtrations for rational G-modules

Abstract

In this paper we investigate Donkin's (p, r)-Filtration Conjecture, and present two proofs of the "if" direction of the statement when p ≥ 2h − 2. One proof involves the investigation of when the tensor product between the Steinberg module and a simple module has a good filtration. One of our main results shows that this holds under suitable requirements on the highest weight of the simple module. The second proof involves recasting Donkin's Conjecture in terms of the identifications of projective indecomposable G r -modules with certain tilting G-modules, and establishing necessary cohomological criteria for the (p, r)-filtration conjecture to hold.

On Tensoring With the Steinberg Representation

“…Although we rely on [KN,Section 8.2] to remove most of the cases, the results in this paper could have been used in other type G 2 cases and lead to very short proofs for the other rank ≤ 2 groups. For example, Theorem 4.2.1 (and its proof) establish the result for SL 2 and SL 3 in all characteristics.…”

On (𝑝,𝑟)-filtrations and tilting modules

“…Andersen [1] showed that (S3) is equivalent to (S4), and proved that both hold when p ≥ 2h − 2. Kildetoft and Nakano [10] gave two alternate proofs that (S4) holds when p ≥ 2h − 2, the second of which came by showing that (S1) implies (S4). As a consequence, it immediately follows that (S4) also holds for SL 3 .…”

On Donkin's Tilting Module Conjecture II: Counterexamples

“…Note that the TMC implies that Hom 1) is a tilting module [KN15]. Moreover, the weights appearing in Hom G 1 (Q 1 (λ), Q 1 (µ)) are less than or equal to 2(p − 1)ρ − λ + ω 0 µ.…”

“…It should be noted that Kildetoft and Nakano, and Sobaje made explicit connections between the TMC and good p-filtrations on G-modules (cf. [KN15], [So18]).…”