Jørgen Vold Rennemo (UiO) - Factorisability in K-theory

Let X be a variety, and (F_n) a sequence of coherent sheaves on the n-th symmetric products Sym^n X. Following Okounkov, we consider the notion of (F_n) being factorisable. Roughly speaking, this means that over points of Sym^(m+n) X corresponding to the union of disjoint configurations in Sym^m X and Sym^n X, the sheaf F_{m+n} is the tensor product of F_m and F_n. Completing a partial argument by Okounkov, we show that if (F_n) is factorisable, then the generating function of Euler characteristics of F_n is the plethystic exponential of a generating function of Euler characteristics of sheaves on X. I'll explain why this is true and why it's useful.