Robinson-Schensted correspondence for unit interval orders

Abstract: Stanley-Stembridge conjecture, currently one of the most famous conjectures in algebraic combinatorics, asks whether a certain generating function with respect to a natural unit interval order is a nonnegative linear combination of complete homogeneous symmetric functions. There are many partial progress on this conjecture, including its connection with the geometry of Hessenberg varieties. Here, instead we study its Schur positivity, which is originally proved by Haiman and Gasharov. We define an analogue of Knuth moves with respect to a natural unit interval order and study its equivalence classes in terms of D graphs introduced by Assaf. Then, we show that if the given order avoids certain two suborders then an analogue of Robinson-Schensted correspondence is well-defined, which proves that the generating function attached to each equivalence class is Schur positive. It is hoped that it proposes a new combinatorial aspect to investigate the Stanley-Stembridge conjectures and cohomology of Hessenberg varieties. This work is joint with Pavlo Pylyavskyy.

algebraic geometryalgebraic topologycombinatoricsrepresentation theory

Audience: researchers in the discipline

T-seminar

Series comments: (Online) seminar on combinatorics, representation theory, symplectic/algebraic geometry related to Torus actions