The Shuffle Theorem, conjectured by Haglund, Haiman, Loehr, Remmel and Ulyanov, and proved by Carlsson and Mellit, describes the characteristic of the Sn-module of diagonal harmonics as a weight generating function over labelled Dyck paths under a line with slope −1. The Shuffle Theorem has been generalized in many different directions, producing a number of theorems and conjectures. We provide a generalized shuffle theorem for paths under any line with negative slope using different methods from previous proofs of the Shuffle Theorem. In particular, our proof relies on showing a ‘stable’ shuffle theorem in the ring of virtual GLℓ-characters. Furthermore, we use our techniques to prove the Extended Delta Conjecture, yet another generalization of the original Shuffle Conjecture.
This video was produced by the Okinawa Institute of Science and Technology, as part of their OIST Representation Theory Seminar series.
