LLT polynomials in the Schiffmann algebra

Abstract: We identify certain combinatorially defined rational functions which, under the shuffle to Schiffmann algebra isomorphism, map to LLT polynomials in any of the distinguished copies \(\Lambda(X^{m,n}) \subset \mathcal{E}\) of the algebra of symmetric functions embedded in the elliptic Hall algebra \(\mathcal{E}\) of Burban and Schiffmann. As a corollary, we deduce an explicit raising operator formula for the \(\nabla\) operator applied to any LLT polynomial. In particular, we obtain a formula for \(\nabla^m s_\lambda\) which serves as a starting point for our proof of the Loehr-Warrington conjecture in a companion paper to this one.

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Date published: Tuesday, April 23, 2024