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  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.

Date published: Wednesday, December 15, 2021