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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 \(\nabla e_{k}\) 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 \(\nabla e_{k}\) 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 \(\nabla ^{m} e_{k}\) 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 \(\nabla ^{m} e_{k}\).

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.