The Delta Conjecture at $q=1$

TL;DR

This paper proves the Delta Conjecture at q=1 by developing combinatorial involutions and bijections, providing explicit expansions and computational tools for symmetric functions related to the Delta operator.

Contribution

It introduces a combinatorial approach using involutions and bijections to prove the Delta Conjecture at q=1, offering new methods for symmetric function analysis.

Findings

Established a combinatorial expansion of _k e_n at q=1.

Proved the Delta Conjecture at q=1 using combinatorial bijections.

Provided structures to compute inner products with symmetric functions.

Abstract

We use a weight-preserving, sign-reversing involution to find a combinatorial expansion of $\Delta_{e_k} e_n$ at $q=1$ in terms of the elementary symmetric function basis. We then use a weight-preserving bijection to prove the Delta Conjecture at $q=1$. The method of proof provides a variety of structures which can compute the inner product of $\Delta_{e_k} e_n|_{q=1}$ with any symmetric function.