A generalization of Kadell's orthogonality ex-conjecture

TL;DR

This paper generalizes Zhou's recursion for Kadell's orthogonality conjecture, extending the understanding of constant term identities related to symmetric functions and $q$-Dyson identities.

Contribution

The paper introduces a new categorization of variables that broadens Zhou's recursion for Kadell's conjecture, advancing the theoretical framework.

Findings

Generalized Zhou's recursion for arbitrary compositions

Extended the closed-form expression for constant terms

Provided a new categorization approach for variables

Abstract

In 2000, Kadell gave an orthogonality conjecture for a symmetric function generalization of the Zeilberger--Bressoud $q$-Dyson constant term identity. The non-zero part of Kadell's conjecture is a constant term identity indexed by a weak composition $v$. This conjecture was first proved by K\'{a}rolyi, Lascoux and Warnaar in 2015. They further formulated a closed-form expression for the above constant term when all parts of the composition $v$ are distinct. In 2021, Zhou obtained a recursion for this constant term for an arbitrary composition $v$. In this paper, by categorizing the variables into two parts, we generalize Zhou's result.