For nonsymmetric Macdonald polynomials Eη(z;q,t)E_\eta(z;q,t), indexed by compositions η\eta, the coefficients Aηλ(r)(q,t)A^{(r)}_{\eta\lambda}(q,t) in the expansion of er(z)Eη(z)e_r(z)E_\eta(z) in the polynomial basis are known only through recursive summation formulas (Baratta's Theorem 7 and Proposition 8). Trial computations show these coefficients almost always factor as products in qq and tt (always for r=1r=1), yet no characterization of when a factorized product form holds is known, and the recursive evaluation of the underlying interpolation-polynomial values Eλi∗(λj)E^*_{\lambda^i}(\lambda^j) leaves open the choice of a reduction index kk that maximizes term vanishing. Both open cores remain unresolved: the problem asks for necessary and sufficient conditions on (η,λ,r)(\eta,\lambda,r) under which Aηλ(r)(q,t)A^{(r)}_{\eta\lambda}(q,t) admits a product form, together with an effective general prescription for the reduction index kk.

Background

A composition η=(η1,…,ηn)\eta=(\eta_1,\dots,\eta_n) is an ordered tuple of nonnegative integers; its degree is ∣η∣=η1+⋯+ηn|\eta|=\eta_1+\cdots+\eta_n. The nonsymmetric Macdonald polynomials Eη(z;q,t)E_\eta(z;q,t), introduced by Cherednik from the double affine Hecke algebra, form a basis of the space of all polynomials in nn variables indexed by compositions, with two deformation parameters qq and tt. They refine the classical symmetric Macdonald polynomials Pκ(z;q,t)P_\kappa(z;q,t) (a basis of symmetric polynomials indexed by partitions κ\kappa, generalizing Schur and Jack polynomials), and they arise as spherical functions and in integrable probability and loop models. A Pieri rule expresses multiplication by an elementary symmetric function er(z)=∑1≤i1<⋯<ir≤nzi1⋯zire_r(z)=\sum_{1\le i_1<\cdots<i_r\le n}z_{i_1}\cdots z_{i_r} in the polynomial basis; for the symmetric family the branching coefficients in er(z)Pκ(z)e_r(z)P_\kappa(z) have been known explicitly since Macdonald's work. For the nonsymmetric family the analogous expansion reads er(z) Eη(z;q−1,t−1)=∑∣λ∣=∣η∣+rAηλ(r)(q,t) Eλ(z;q−1,t−1),e_r(z)\,E_\eta(z;q^{-1},t^{-1})=\sum_{|\lambda|=|\eta|+r}A^{(r)}_{\eta\lambda}(q,t)\,E_\lambda(z;q^{-1},t^{-1}), and the case r=1r=1 was resolved in closed form by Baratta (2009), building on the Jack-polynomial precedent of Forrester and McAnally (2004), where the r=1r=1 coefficients were found to be simple products and a unit coefficient was identified. Baratta's 2011/2012 paper extends this to general rr. The derivation uses the interpolation Macdonald polynomials Eη∗(z;q,t)E^*_\eta(z;q,t), inhomogeneous polynomials whose top homogeneous component is Eη(z;q−1,t−1)E_\eta(z;q^{-1},t^{-1}) and which satisfy extra vanishing properties (Eη∗(μ)=0E^*_\eta(\mu)=0 for all μ\mu of degree ∣μ∣≤∣η∣|\mu|\le|\eta| with μ≠η\mu\ne\eta). The coefficients Aηλ(r)A^{(r)}_{\eta\lambda} vanish unless η⪯′λ\eta\preceq'\lambda, where ⪯′\preceq' is the partial order on compositions defined by η⪯′λ\eta\preceq'\lambda iff there exists a permutation σ\sigma of {1,…,n}\{1,\dots,n\} with ηi<λσ(i)\eta_i<\lambda_{\sigma(i)} whenever i<σ(i)i<\sigma(i) and ηi≤λσ(i)\eta_i\le\lambda_{\sigma(i)} whenever i≥σ(i)i\ge\sigma(i) (in particular ∣λ∣≥∣η∣|\lambda|\ge|\eta|), and are obtained by a triangular recursion (Theorem 7) in terms of evaluations Eλi∗(λj)E^*_{\lambda^i}(\lambda^j) between the intermediate compositions λi\lambda^i with ∣λi∣=∣η∣+i|\lambda^i|=|\eta|+i and η⪯′λi⪯′λ\eta\preceq'\lambda^i\preceq'\lambda that arise along chains of successors in the recursion. Proposition 8 reduces such evaluations: with Φ\Phi the raising operator Φ(η1,…,ηn)=(η2,…,ηn,η1+1)\Phi(\eta_1,\dots,\eta_n)=(\eta_2,\dots,\eta_n,\eta_1+1) cyclically moving the first component to the last position with increment, sjs_j the operator swapping components jj and j+1j+1, and, for each subset Ik⊆{1,…,n−1}I_k\subseteq\{1,\dots,n-1\}, DIk(η)=σk⋯σn−1 Φ σ1⋯σk−1(η)D_{I_k}(\eta)=\sigma_k\cdots\sigma_{n-1}\,\Phi\,\sigma_1\cdots\sigma_{k-1}(\eta) where σj=sj\sigma_j=s_j if j∈Ikj\in I_k and σj\sigma_j is the identity otherwise, the set Dk(λi)={ν:ν=DIk(λi) for some Ik}D_k(\lambda^i)=\{\nu:\nu=D_{I_k}(\lambda^i)\text{ for some }I_k\} carries a summation identity for Eλi∗(λj)E^*_{\lambda^i}(\lambda^j); the only constraint on kk is that Dk(λi)D_k(\lambda^i) contain some ν⪯′λj\nu\preceq'\lambda^j. The same machinery evaluates the generalized binomial coefficients (ην)q,t=Eη∗(ν)/Eν∗(ν)\binom{\eta}{\nu}_{q,t}=E^*_\eta(\nu)/E^*_\nu(\nu) introduced by Sahi, for which no explicit general formula is known beyond Baratta's summation expressions. The resulting recursive formulas for Aηλ(r)A^{(r)}_{\eta\lambda} are combinatorially complex summations. Trial computations show that these coefficients can often be simplified to factorized products of terms such as qatb−1q^a t^b-1, and for r=1r=1 this factorization always holds; but the recursion itself gives no criterion for when factorization occurs, and the choice of the index kk controlling the efficiency of the reduction has no known effective prescription.

Problem Statement

For the nonsymmetric Macdonald polynomials Eη(z;q,t)E_\eta(z;q,t) in nn variables, consider the Pieri-type coefficients Aηλ(r)(q,t)A^{(r)}_{\eta\lambda}(q,t) defined by the expansion er(z)Eη(z;q−1,t−1)=∑∣λ∣=∣η∣+rAηλ(r)(q,t)Eλ(z;q−1,t−1)e_r(z)E_\eta(z;q^{-1},t^{-1})=\sum_{|\lambda|=|\eta|+r}A^{(r)}_{\eta\lambda}(q,t)E_\lambda(z;q^{-1},t^{-1}), currently computable through the recursion of Baratta's Theorem 7 together with the reduction of Proposition 8. The overarching objective is to turn these recursive summation formulas into explicit, structured, efficiently computable formulas for the coefficients (and, as a direct corollary, for the generalized binomial coefficients (ην)q,t\binom{\eta}{\nu}_{q,t}). Two precisely stated questions from the source paper constitute the open core. (1) Product forms: give necessary and sufficient conditions on the compositions η⪯′λ\eta\preceq'\lambda (with ∣λ∣=∣η∣+r|\lambda|=|\eta|+r) and on r≥2r\ge2 under which Aηλ(r)(q,t)A^{(r)}_{\eta\lambda}(q,t) admits a factorized product form — that is, equals a monomial in qq and tt times a finite product of integer powers of factors of the form (qatb−1)(q^a t^b-1) with a,b∈Za,b\in\mathbb{Z}, as always realized by the closed r=1r=1 formulas and as observed empirically for r=2r=2. (2) Reduction index: give an effective, general prescription for choosing the index kk in the reduction of Proposition 8 — that is, a structural rule that, given λi⪯′λj\lambda^i\preceq'\lambda^j with i<j−1i<j-1, determines k∈{1,…,n}k\in\{1,\dots,n\} directly from the compositions λi\lambda^i and λj\lambda^j, without exhaustively enumerating the sets Dk(λi)D_k(\lambda^i) for all candidate values of kk (brute-force search over the finite range of kk is always available and does not resolve the open problem), such that, over the values of kk for which the reduction of Proposition 8 is non-trivial (that is, those for which Dk(λi)D_k(\lambda^i) contains some ν⪯′λj\nu\preceq'\lambda^j), the number of ν∈Dk(λi)\nu\in D_k(\lambda^i) with ν̸⪯′λj\nu\not\preceq'\lambda^j is maximized — with a proof that the prescription achieves this optimality. The two questions serve the single objective of explicit, efficient Pieri-type formulas: the first concerns the structure of the output coefficients, the second the cost of the computation that produces them.

References

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