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On tensoring with the Steinberg representation. (English) Zbl 07271116
Summary: 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 $$r$$ th Steinberg module with a simple module with $$p^r$$ th restricted highest weight admits a good filtration. In this paper we verify this statement (i) when $$p \geq 2h - 4$$ ($$h$$ is the Coxeter number), (ii) for all rank two groups, (iii) for $$p \geq 3$$ 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.

##### MSC:
 20G05 Representation theory for linear algebraic groups 20G10 Cohomology theory for linear algebraic groups 17B10 Representations of Lie algebras and Lie superalgebras, algebraic theory (weights) 20G15 Linear algebraic groups over arbitrary fields
LiE; Magma
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