A family of q-Dyson style constant term identities

Abstract

By generalizing Gessel-Xin's Laurent series method for proving the Zeilberger-Bressoud q-Dyson Theorem, we establish a family of qDyson 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, C).

Constant term identities and Poincaré polynomials

“…In Section 4, we apply Theorem 1.2 to prove and generalise Kadell's orthogonality conjecture. Finally, in Section 5, answering a question raised by the anonymous referee, we show that Kadell's orthogonality conjecture implies a conjecture of Sills [32] proved previously by Lv, Xin and Zhou using different means [28].…”

Two Coefficients of the Dyson Product

“…Using this algorithm, Sills [8] guessed and proved closed-form expressions for M to be xs xr , xsxt x 2 r and xtxu xrxs . These results and their q-analogies were recently generalized for M with a square free numerator by Lv, Xin and Zhou [7] by extending Gessel-Xin's Laurent series method [3] for proving the q-Dyson Theorem.…”

A Laurent Series Proof of the Habsieger-Kadell $q$-Morris Identity

“…We will follow notations in [9,16], where different versions of the vanishing lemma were proposed for dealing with q-Dyson related constant terms. The new vanishing lemma will be handled by the same idea but we have to carry out the details.…”

“…Note that this basic idea was used by Habsieger for q-Selberg integral in [12]. The equality at the d vanishing points are not hard to handle by the techniques in [9,16]. But in this approach, we have to deal with two problems: i) the multiple roots problem for small k; ii) the d + 1-st suitable point is hard to find.…”