UC Berkeley

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 Λ (X m, n) ⊂ E of the algebra of symmetric functions embedded in the elliptic Hall algebra ϵ of Burban and Schiffmann. As a corollary, we deduce an explicit raising operator formula for the ∇ operator applied to any LLT polynomial. In particular, we obtain a formula for ∇m s λ which serves as a starting point for our proof of the Loehr-Warrington conjecture in a companion paper to this one.