The Stanley-Stembridge conjecture was a conjecture in algebraic combinatorics about the nonnegativity of the coefficients of certain symmetric functions when expressed in terms of elementary symmetric functions. The original form of the conjecture, as formulated by Stanley and Stembridge in their 1993 paper, was proved by Tasuyuki Hikita in 2024, and a stronger positivity conjecture, formulated by Shareshian and Wachs, was proved by OpenAI on October 6, 2026.   

The original motivation for the conjecture had to do with positivity properties of immanants, basically determinants of n x n matrices where the sign function is replaced by a character of the symmetric group. But very quickly, Haiman realized that the conjecture was connected to objects in geometric representation theory associated to the general linear group. Later Shareshian and Wachs formulated a relationship between the symmetric functions in the conjecture and certain subvarieties of the flag variety known as Hessenberg varieties. That this relationship holds was proved in 2015 by me and Tim Chow, and independently by Guay-Paquet. Using this, you can think of Stanley-Stembridge as a conjecture about the monodromy action on the total cohomology of a family of Hessenberg varieties parametrized by n x n matrices with n distinct eigenvalues. The Shareshian-Wachs positivity conjecture proved by OpenAI is then a similar statement in each cohomological degree.

This leaves us in the following state: we know the original conjecture and even the positivity in each degree conjectured by Shareshian and Wachs. But we do not know a stronger conjecture made by Haiman. Also, while Hessenberg varieties make sense for arbitrary reductive groups, such as the orthogonal or symplectic groups, we still do not have an analogous positivity conjecture or a fast and general way to compute the monodromy action.  

I'll explain this history in more detail along with my work with Hong and Lee whose main goal is to understand the situation for arbitrary reductive groups.