A combinatorial skewing formula for the Rise Delta Theorem

TL;DR

This paper establishes a new combinatorial proof connecting symmetric functions related to the Delta Conjecture and the Rational Shuffle Theorem, expanding understanding of these algebraic structures.

Contribution

It introduces a combinatorial proof of a skewing identity that links the Delta and Rational Shuffle Theorems, generalizing previous formulas.

Findings

Proves a skewing formula for the Rise Delta Theorem

Provides a purely combinatorial proof of the skewing identity

Connects symmetric functions in the Delta and Rational Shuffle Theorems

Abstract

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 Rational 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 Rational Shuffle Theorem.