Combinatorial Theory

We prove that the symmetric function \(\Delta'_{e_{k-1}}e_n\) appearing in the Delta Conjecture can be obtained from the symmetric function in the Rectangular Shuffle Theorem by applying a Schur skewing operator. This generalizes a formula by the first and third authors for the Delta Conjecture at \(t=0\), and follows from work of Blasiak, Haiman, Morse, Pun, and Seelinger. Our main result is that we also provide a purely combinatorial proof of this skewing identity, giving a new proof of the Rise Delta Theorem from the Rectangular Shuffle Theorem.

Mathematics Subject Classifications: 05E05

Keywords: Delta conjecture, parking functions, sign reversing involutions, skewing formula, Rectangular shuffle theorem, dinv, symmetric functions