2018/04/02 by Bendel, Christopher P., Nakano, Daniel K., Pillen, Cornelius +1
#18G05 #20G05 #20J06 #FOS: Mathematics #Group Theory (math.GR) #Representation Theory (math.RT)
paper · doi:10.48550/arxiv.1804.00613
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 prth 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.