The nonsymmetric compositional Delta theorem

TL;DR

This paper generalizes the compositional Delta theorem to a nonsymmetric setting, introduces nonsymmetric operators, and connects these to flagged LLT polynomials, providing new identities and conjectures.

Contribution

It extends the symmetric compositional Delta theorem to nonsymmetric cases, introduces nonsymmetric operators, and links these to flagged LLT polynomials.

Findings

Derived nonsymmetric identities evaluated in terms of flagged LLT polynomials.

Introduced nonsymmetric variants of the $ abla$ and $ au^*$ operators.

Recovered original theorem via Weyl symmetrization of nonsymmetric identities.

Abstract

Extending the symmetric framework of D'Adderio and Mellit, we establish a nonsymmetric generalization of the compositional Delta theorem. Building on Blasiak et al.'s theory of flagged LLT polynomials, we derive signed and unsigned nonsymmetric identities evaluated in terms of flagged LLT polynomials. Furthermore, by introducing nonsymmetric variants of the $\nabla$ and $\tau^*$ operators, we obtain a novel operator formulation. We show that applying Weyl symmetrization to these nonsymmetric identities systematically recovers the original compositional Delta theorem. Finally, we propose analogous conjectures regarding stable atom positivity.