LLT polynomials in the Schiffmann algebra

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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mi mathvariant="normal">Λ</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:msup> <m:mi>X</m:mi> <m:mrow> <m:mi>m</m:mi> <m:mo>,</m:mo> <m:mi>n</m:mi> </m:mrow> </m:msup> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo>⊂</m:mo> <m:mi mathvariant="script">E</m:mi> </m:mrow> </m:math> Λ(X^{m{,}n}) of the algebra of symmetric functions embedded in the elliptic Hall algebra ℰ 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:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msup> <m:mo>∇</m:mo> <m:mi>m</m:mi> </m:msup> <m:msub> <m:mi>s</m:mi> <m:mi>λ</m:mi> </m:msub> </m:mrow> </m:math> ∇ᵐsλ which serves as a starting point for our proof of the Loehr–Warrington conjecture in a companion paper to this one.