James's conjecture holds for weight four blocks of Iwahori-Hecke algebras. (Q2466937)

James's conjecture claims that in certain cases, the decomposition numbers for the Iwahori-Hecke algebra of the symmetric group over a field of positive characteristic coincide with the decomposition numbers for a corresponding Iwahori-Hecke algebra defined over \(\mathbb{C}\). The latter can be computed by using the Ariki-LLT theory. This conjecture still remained open up to now, though it has been verified in a number of special cases. In this paper, the author verifies this conjecture for all the weight four blocks of the Iwahori-Hecke algebras, generalizing his earlier work for all the weight three blocks.