Let $\Lambda$ be a preprojective algebra of type $A,D,E$, and let $G$ be the corresponding semisimple simply connected complex algebraic group. We study rigid modules in subcategories $\mathrm{Sub}Q$ for $Q$ an injective $\Lambda$-module, and we introduce a mutation operation between complete rigid modules in $\mathrm{Sub}Q$. This yields cluster algebra structures on the coordinate rings of the partial flag varieties attached to $G$.

We first note that a result of Gowers on product-free sets in groups has an unexpected consequence: If $k$ is the minimal degree of a representation of the finite group $G$, then for every subset $B$ of $G$ with $\left|B\right|>\left|G\right|/k^{1/3}$ we have $B^3=G$. We use this to obtain improved versions of recent deep theorems of Helfgott and of Shalev concerning product decompositions of finite simple groups, with much simpler proofs. On the other hand, we prove a version of Jordan’s theorem which implies that if $k\ge 2$, then $G$ has a proper subgroup...