A Family of $q$-Dyson Style Constant Term Identities

TL;DR

This paper generalizes the $q$-Dyson constant term identities by extending Laurent series methods, providing explicit formulas for coefficients, and encompassing several conjectures and character formulas in a unified framework.

Contribution

It introduces a new family of $q$-Dyson identities that unify and extend previous conjectures and formulas using a generalized Laurent series approach.

Findings

Established a family of explicit $q$-Dyson constant term identities.

Unified several conjectures of Sills as special cases.

Extended Stembridge's formulas for $SL(n,\mathbb{C})$ characters.

Abstract

By generalizing Gessel-Xin's Laurent series method for proving the Zeilberger-Bressoud $q$-Dyson Theorem, we establish a family of $q$-Dyson style constant term identities. These identities give explicit formulas for certain coefficients of the $q$-Dyson product, including three conjectures of Sills' as special cases and generalizing Stembridge's first layer formulas for characters of $SL(n,\mathbb{C})$.