On the $q$-Dyson orthogonality problem

Abstract

By combining the Gessel-Xin method with plethystic substitutions, we obtain a recursion for a symmetric function generalization of the q-Dyson constant term identity also known as the Zeilberger-Bressoud q-Dyson theorem. This yields a constant term identity which generalizes the non-zero part of Kadell's orthogonality ex-conjecture and a result of Károlyi, Lascoux and Warnaar.

A recursion for a symmetric function generalization of the $q$-Dyson constant term identity

“…, u n ) of integers we denote by u + the sequence obtained from u by ordering the u i in weakly decreasing order (so that u + is a partition if u is a composition), then Károlyi et al also proved a closed-form expression for D v,v + (a) in the case when v is a composition all of whose parts are distinct, i.e., v i = v j for all 1 ≤ i < j ≤ n. Subsequently, Cai [4] gave an inductive proof of Kadell's conjecture. Recently, we [19] obtained a recursion for D v,v + (a) if v is a composition such that its largest part has multiplicity one, see Corollary 1.2 below.…”

“…, 0) the constant term D v,v + (a) corresponds to the q-Dyson constant term (the left-hand side of (1.1)). Using the recursion (1.5) and the q-Dyson identity (1.1), we can obtain a closed-form formula for D v,v + (a) for arbitrary compositions v. If the largest part of v has multiplicity one in v, then Theorem 1.1 reduces to [19,Theorem 1.3].…”

A symmetric function generalization of the Zeilberger--Bressoud $q$-Dyson theorem

“…Károlyi et al also proved a closed-form expression for D v,v + (a) in the case when v is a weak composition all of whose parts are distinct, i.e., v i = v j for all 1 ≤ i < j ≤ n. Subsequently, Cai [3] gave an inductive proof of Kadell's conjecture. Recently, we got a recursion for D v,v + (a) if v is a weak composition with unique largest part [12]. Later, we obtained a recursion for D v,v + (a) for arbitrary nonzero weak composition v in [13].…”