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A Shuffle Theorem for Paths Under Any Line

Mark Haiman

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

More generally, Haglund et al. conjectured an identity giving ${\mathrm{\nabla }}^m{e}_k$ 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 $(e_k[-M{X}^{m,n}]\cdot 1)$ 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 $e_k[-M{X}^{m,n}]$ is an element of Schiffmann's elliptic Hall algebra, acting as an operator on symmetric functions such that for $n=1$, the expression $(e_k[-M{X}^{m,n}]\cdot 1)$ reduces to ${\mathrm{\nabla }}^m{e}_k$.

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, Jennifer Morse, Anna Pun and George Seelinger.