Intertwining operators and polynomials associated with the symmetric group

Abstract

There is an algebra of commutative differential-difference operators which is very useful in studying analytic structures invariant under permutation of coordinates. This algebra is generated by the Dunkl operators\(T_i : = \frac{\partial }{{\partial x_i }} + k\sum\nolimits_{j \ne i} {\frac{{1 - (ij)}}{{x_i - x_j }}} \), (i=1, ...,N, where (ij) denotes the transposition of the variablesxixj andk is a fixed parameter). We introduce a family of functions {pα}, indexed bym-tuples of non-negative integers α = (α1, ..., αm) form≤N, which allow a workable treatment of important constructions such as the intertwining operatorV. This is a linear map on polynomials, preserving the degree of homogeneity, for which\(T_i V = V\frac{\partial }{{\partial x_i }}\),i = 1, ...,N, normalized byV1=1 (seeDunkl, Canadian J. Math.43 (1991), 1213–1227). We show thatTipα=0 fori>m, and $$V(x_1^{\alpha _1 } \cdots x_m^{\alpha _m } ) = \frac{{\lambda _1 !\lambda _2 ! \cdots \lambda _m !}}{{\left( {Nk + 1} \right)_{\lambda _1 } \left( {Nk - k + 1} \right)_{\lambda _2 } \cdots (Nk - (m - 1)k + 1)_{\lambda _m } }}p_\alpha + \sum\limits_\beta {A_{\beta \alpha } p_{\beta ,} } $$ where (λ1, λ2, ..., λm) is the partition whose parts are the entries of α (That is, λ1➮ λ2➮ ... λm➮0), β = (β1, ..., βm), ∑i=1m βi = ∑i=1m αm and the sorting of β is a partition strictly larger than λ in the dominance order. This triangular matrix representation ofV allows a detailed study. There is an inner product structure on span {pα} and a convenient set of self-adjoint operators, namelyTiρi, whereρipα ≔p(α1, ...., αi + 1, ..., αm). This structure has a bi-orthogonal relationship with the Jack polynomials inm variables. Values ofk for whichV fails to exist are called singular values and were studied byDe Jeu, Opdam, andDunkl in Trans. Amer. Math. Soc.346 (1994), 237–256. As a partial verification of a conjecture made in that paper, we construct, for anya=1,2,3,... such that gcd(N−m+1,a)<(N−m+1)/m andm≤N/2, a space of polynomials annihilated by eachTi fork=−a/(N−m+1) and on which the symmetric groupSN acts according to the representation (N−m, m).