Representations of the Double Dyck Path Algebra and Beyond

The shuffle theorem is a celebrated result in algebraic combinatorics that identifies three objects: the Frobenius character of certain $S_n$ representations, the action of the elliptic Hall algebra on symmetric functions, and a particular combinatorial expression in terms of labeled lattice paths (known as parking functions). In the original proof given by Carlsson and Mellit, this theorem was established through the introduction of the so-called/Double Dyck Path Algebra./ A crucial step in their proof involved the /polynomial representation /of this algebra.

I will explain joint work with Eugene Gorsky and Jose Simental, in which we construct new families of /calibrated/ representations of the double Dyck path algebra on graded posets and provide a completely combinatorial classification of these representations. By showing that any Dyck path  element in this algebra can be "reduced to level one", we also prove that the elliptic Hall algebra  arises as a specific subalgebra. This result opens the door to new, unexplored "shuffle theorems" and associated Catalan combinatorics that emerge by replacing the polynomial representation in the original Shuffle Theorem with our new calibrated representations.