Equality conditions for correlation inequalities

We prove equality conditions for the Ahlswede–Daykin (AD) inequality and the Fortuin–Kasteleyn–Ginibre (FKG) inequality. We then present a number of applications and special cases of these equality conditions. These include Björner’s and Fishburn’s inequalities for linear extensions of finite posets, the Lam–Postnikov–Pylyavskyy (LPP) and the Okounkov inequalities for Schur positivity of products of Schur functions. We conclude with equality conditions for the Ahlswede–Daykin–Schur (ADS) inequality recently introduced in Chan–Chen–Pak–Soskin (2026), which is an AD type extension of the LPP inequality.