More generally, Haglund et al. conjectured an identity giving $\nabla^m e_k$ as a sum over LLT polynomials indexed by paths under the line segment from $(0, k)$ to $(km, 0)$. Mellit later proved a generalization of this conjecture by Bergeron, Garsia, Sergel Leven and Xin, which gives $e_k[-MX^{m,n}]\cdot 1$ as a sum over paths under the segment from $(0, kn)$ to $(km, 0)$ for any pair of positive integers expressed in the form $km, kn$ with $m, n$ coprime. Here $e_k[-MX^{m,n}]$ is an element of Schiffmann's elliptic Hall algebra, acting as an operator on symmetric functions such that for $n=1$, the expression $e_k[-MX^{m,n}]\cdot 1$ reduces to $\nabla^m e_k$.

We show that the shuffle theorem has a natural further extension involving lattice paths under any line segment between real points $(0,s)$ and $(r,0)$ on the positive axes, reducing to the Bergeron et al. and Mellit shuffle theorem when $(r,s)=(km,kn)$ are integers.

Our proof uses a different method than the proofs of previous versions of the shuffle theorem, and is surprisingly simple. This is joint work with Jonah Blasiak, Mark Haiman, Jennifer Morse, and George Seelinger.

### CAGE: Philadelphia Area Combinatorics and Alg. Geometry Seminar

Thursday, February 25, 2021 - 3:30pm

#### Anna Pun

University of Virginia

The shuffle theorem, conjectured by Haglund, Haiman, Loehr, Remmel and Ulyanov, and proved by Carlsson and Mellit, is a combinatorial identity expressing the symmetric function $\nabla e_k$ as a sum over LLT polynomials indexed by Dyck paths, that is, lattice paths lying under the linne segment from $(0,k)$ to $(k,0)$. The function $\nabla e_k$ arises in the theory of Macdonald polynomials and gives the doubly graded character of the ring of diagonal coinvariants for the symmetric group $S_k$.