A raising operator formula for $\nabla$ on an LLT polynomial

Abstract: The symmetric function operator $\nabla$ arose in the theory of Macdonald polynomials and its action on various bases has been the subject of numerous conjectures over the last two decades. It developed that $\nabla$ is but a shadow of a more complete picture involving the elliptic Hall algebra of Burban and Schiffmann. This algebra is generated by subalgebras $\Lambda(X^{m,n})$ isomorphic to the ring of symmetric functions, one for each coprime pair of integers $(m,n)$. We identify certain combinatorially defined rational functions which correspond to LLT polynomials in any of the subalgebras $\Lambda(X^{m,n})$. As a corollary, we deduce an explicit raising operator formula for $\nabla$ on any LLT polynomial. This is joint work with Mark Haiman, Jennifer Morse, Anna Pun, and George Seelinger.

combinatorics

Audience: researchers in the topic

York University Applied Algebra Seminar