Title: Chern shadows and plethystic recursions

Abstract: Irreducible (polynomial) representations of GL(r) are indexed by partitions, and are known as Schur modules - given any representation, it is in general difficult to decompose it as sum of irreducible ones. A particularly important case is the one of a plethysm: decompose the composition of two Schur modules as sum of other Schur modules. For example, Sym^2(Sym^2(V)) decomposes in Sym^4V+S^(2,2)V as a GL(V)-module. Finding explicit formulas for plethysms is one of the most difficult problems in the combinatorics of symmetriic functions.

In this joint work with A. D'Andrea and E. Fatighenti we introduce a combinatorial object, that we call Chern shadow, to any Schur module and show that it satisfies pleasing properties. As an application we deduce a recursive formula to compute a given plethysm. Time permitting, I would explain the geometric problem that led to this work.