An Elliptic Hypergeometric Function Approach to Branching Rules

Abstract

We prove Macdonald-type deformations of a number of well-known classical branching rules by employing identities for elliptic hypergeometric integrals and series. We also propose some conjectural branching rules and allied conjectures exhibiting a novel type of vanishing behaviour involving partitions with empty 2-cores.

Proof of some Littlewood identities conjectured by Lee, Rains and Warnaar

“…. By [14,Lemma 2.3] we may alternatively express this as (4.3) q b(λ) C e λ ′ (q 2m ; q)H o λ ′ (q) C o λ ′ (q 2m ; q)H e λ ′ (q) = q b(λ ′ ) C e λ (q −2m ; q)H o λ (q) C o λ (q −2m ; q)H e λ (q) This establishes (4.1). For (4.2) the same procedure applies with the substitution (t 0 , t 1 , t 2 , t 3 ) = (1, −1, q, −q) and by using the integral (3.3).…”

“…which is Gustafson's generalised Askey-Wilson integral [9]. The virtual Koornwinder integral can be evaluated for many choices of the argument f , see [14,22,23,24]. In particular, the vanishing integrals of the next section may be expressed in terms of virtual Koornwinder integrals.…”

“…This, however, does not seem to shed light on a more explicit expression for the evaluation of these sums. In particular, the specialisations of K (m n ) above are not contained in [14,Lemma 4.1].…”

“…In Section 3 we prove a pair of vanishing integrals for Schur functions again conjectured by Lee, Rains and Warnaar in the Macdonald case [14, Conjecture 9.2]. Then, in Section 4, we follow the techniques of [24] to prove the bounded analogues of Theorem 1.1 conjectured in [14,Conjecture 9.4]. The theorem then follows by taking an appropriate limit.…”

“…The purpose of this note is to prove the Schur function case of a pair of Littlewood identities for Macdonald polynomials recently conjectured by Lee, Rains and Warnaar [14,Conjecture 9.5]. To state these we need some notation.…”