Exchange identities and symmetric slices of the valley Delta conjecture

Abstract

The valley Delta conjecture of Haglund, Remmel and Wilson predicts that the symmetric function $Δ'_{e_{n-k-1}} e_n$ equals a generating function over labelled Dyck paths with $k$ decorated contractible valleys. Unlike the rise version, which is now a theorem, the valley version remains open; indeed it is not even known that its combinatorial side is symmetric. We prove that the coefficients of $t^0$ and $t^1$ in the valley generating function are symmetric functions for all $n\ge 1$ and $k\ge 0$; equivalently, the fixed-diagonal-multiset slices of area at most one are symmetric. The theorem follows from an adjacent exchange identity for scaffold classes, which we prove in a strictly stronger form refined by the numbers of undecorated rows carrying the labels $r,r+1$ between consecutive rows with other labels. The proof develops a transfer-operator calculus in a $q$-deformed two-variable algebra generated by two commuting half-twists. In this algebra the exchange reduces to two scalar symmetric-series identities for the operator $T=u\mathfrak{d}+v$. We also verify the refined identity computationally at area two over extensive finite ranges and state the resulting general conjecture.