On Tensoring With the Steinberg Representation

Abstract

Let G be a simple, simply connected algebraic group over an algebraically closed field of prime characteristic p > 0. Recent work of Kildetoft and Nakano and of Sobaje has shown close connections between two long-standing conjectures of Donkin: one on tilting modules and the lifting of projective modules for Frobenius kernels of G and another on the existence of certain filtrations of G-modules. A key question related to these conjectures is whether the tensor product of the rth Steinberg module with a simple module with p r th restricted highest weight admits a good filtration. In this paper we verify this statement when (i) p ≥ 2h − 4 (h is the Coxeter number), (ii) for all rank two groups, (iii) for p ≥ 3 when the simple module corresponds to a fundamental weight and (iv) for a number of cases when the rank is less than or equal to five.

On Donkin's Tilting Module Conjecture II: Counterexamples

“…Jantzen's Question. It was shown in [BNPS20a,5.6] that Conjecture 2.3.3 holds in this case. Moreover, from [So18] we know that an affirmative answer to Jantzen's Question (i.e., Question 2.3.1) together with Corollary 2.3.3 implies that the Tilting Module Conjecture holds.…”

On Donkin's Tilting Module Conjecture I: Lowering the Prime

“…The hypothesis that St r ⊗L(µ) is tilting for all restricted weights µ is known to hold in a number of cases (cf. [BNPS19]). Suppose that (b) holds.…”

“…In subsequent sections this fact will be used when we verify the TMC for other low rank cases. For Φ = A n , n ≤ 3, by [KN15] or [BNPS19], it is known that St 1 ⊗L(µ) admits a good filtration for all µ ∈ X 1 . Therefore, by Theorem 4.4.1, in order to verify the TMC in these cases it suffices to prove that ∇( λ) has a good p-filtration for all λ ∈ X 1 .…”

Counterexamples to the Tilting and $(p,r)$-Filtration Conjectures

“…In a recent paper, the authors lowered the bound to p ≥ 2h − 4 (cf. [BNPS18]). For rank 2 groups (including G 2 ), Conjecture 2.2.3(⇒) was proved for all p in [KN15] and [BNPS18].…”

“…[BNPS18]). For rank 2 groups (including G 2 ), Conjecture 2.2.3(⇒) was proved for all p in [KN15] and [BNPS18].…”