Let G be a simple algebraic group split over R, for instance G = SL(n,R). Generalizing the classical notion of totally positive matrices, Lusztig introduced in the 1990's the subset of totally positive (resp. totally nonnegative) elements of G. In 1998 he extended this notion to the partial flag manifolds G/P, for example the Grassmannians.

One combinatorial problem arising from this is to find optimal criteria for an element of G (or G/P) to be totally positive (resp. totally nonnegative). In 2001, Fomin and Zelevinsky invented the notion of a cluster algebra, motivated in part by this combinatorial problem which they solved completely in the case of G.

After reviewing this story, I will outline some recent progress in the case of G/P.