Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A generalization of Kadell's orthogonality ex-conjecture

Zihao Huang, Wenlong Jiang, Yue Zhou

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 Károlyi, 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.

A generalization of Kadell's orthogonality ex-conjecture

Abstract

In 2000, Kadell gave an orthogonality conjecture for a symmetric function generalization of the Zeilberger--Bressoud -Dyson constant term identity. The non-zero part of Kadell's conjecture is a constant term identity indexed by a weak composition . This conjecture was first proved by Károlyi, Lascoux and Warnaar in 2015. They further formulated a closed-form expression for the above constant term when all parts of the composition are distinct. In 2021, Zhou obtained a recursion for this constant term for an arbitrary composition . In this paper, by categorizing the variables into two parts, we generalize Zhou's result.
Paper Structure (6 sections, 11 theorems, 80 equations)

This paper contains 6 sections, 11 theorems, 80 equations.

Table of Contents

  1. Introduction
  2. A splitting formula for $F_{n,n_0}(a;x,w)$
  3. Proof of Theorem \ref{['Thm-van']}
  4. An inductive formula for $D_{v,\lambda}(a;n,n_0)$
  5. Proof of Theorem \ref{['p-recu']}
  6. Other forms of Theorem \ref{['Thm-van']} and Corollary \ref{['Thm-rec']}

Key Result

Theorem 1.1

Let $v \in \mathbb{Z}^{n}$ and $\lambda$ be a partition such that $|v|=|\lambda|$. For $I=\{i_1,\ldots,i_j\}\subseteq \{1,\ldots,n\}$, denote $p_I:=\:|\: \{ i_k : i_k\leq n_0, 1\leq k\leq j\} \:|\:$. Then $D_{v,\lambda}(a;n,n_0)=0$, if there exists an integer $0<j<n$, such that holds for every $j$-elements subsets $I=\{i_1,\ldots,i_j\}\subseteq \{1,\ldots,n\}$.

Theorems & Definitions (19)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Lemma 2.1
  • proof
  • Proposition 2.2
  • proof
  • proof : Proof of Theorem \ref{['Thm-van']}
  • Proposition 4.1
  • ...and 9 more