Georgia Institute of Technology College of Sciences

The Ahlswede–Daykin inequality (also known as the Four Function Theorem) is a classical cornerstone of probability theory and statistical mechanics. Its power lies in its ability to reduce proving correlation and log-concave inequalities to a set of local, usually straightforward, conditions. In this talk, we extend this classical result to the setting of symmetric functions, introducing a new framework tailored for proving Schur positivity. As an application, we give a simple proof of the Schur log-concave inequality for Schur polynomials, a landmark result originally established by Lam–Postnikov–Pylyavskyy (2007). We also establish Schur log-concave inequality for Grothendieck polynomials, thereby resolving a conjecture of Mihalcea.

This talk is a joint work with Hong Chen, Igor Pak, and Daniel Soskin.