Factorisability in K-theory

We revisit and give a detailed proof of a lemma of Okounkov showing that, for a scheme X with a torus action, the Euler characteristic generating function associated with a “factorisable” sequence of torus-equivariant coherent sheaves on the symmetric powers ${\mathrm{Sym}}^{n}X$ equals the plethystic exponential of the generating function of Euler characteristics of some sequence of sheaves on X.