The Delta Conjecture at 𝑞=1

Abstract

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