On character formulas for simple and tilting modules

We show that the characters of tilting modules can be used, in a concrete and explicit way, to obtain the simple characters of a connected reductive algebraic group G over an algebraically closed field k of characteristic p, for all p. Thus, once an algorithm for computing the characters of the indecomposable tilting G-modules has been found, an algorithm for the simple characters has been also. An immediate implication is that work by Achar, Makisumi, Riche, and Williamson can be used to obtain the characters for simple G-modules when p>h, the Coxeter number of G.