Intertwining operators and polynomials associated with the symmetric group
Intertwining operators and polynomials associated with the symmetric group
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与对称群相关的交织运算符和多项式
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发表时间：
1998
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通讯作者：
C. Dunkl
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文献类型：
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作者：
C. Dunkl
摘要交换微分-差分算子是研究解析结构在坐标置换下不变的重要代数。这个代数由Dunkl算子生成
$$T_i : = frac{partial }{{partial x_i }} + ksum
olimits_{j
e i} {frac{{1 - (ij)}}{{x_i - x_j }}} $$
， （i=1，…），N，其中（ij）为变量的变换（xixj， k为固定参数）。我们引入了一个函数{族}pα，以非负整数α = (α1，…， αm)形式≤N，它允许对重要结构（如缠结算子v）进行可行的处理。这是多项式上的线性映射，保持了同质性，对于它
$$T_i V = Vfrac{partial }{{partial x_i }}$$
，i = 1，…[j] .数学学报，2003(1):1 - 2。我们证明tip α=0对于>m，并且
$$V(x_1^{alpha _1 } cdots x_m^{alpha _m } ) = frac{{lambda _1 !lambda _2 ! cdots lambda _m !}}{{left( {Nk + 1}
ight)_{lambda _1 } left( {Nk - k + 1}
ight)_{lambda _2 } cdots (Nk - (m - 1)k + 1)_{lambda _m } }}p_alpha + sumlimits_eta {A_{eta alpha } p_{eta ,} } $$
式中(λ1, λ2，…， λm)为分区，其各部分为α的项(即λ1➮λ2➮…λm➮0),β = (β1，…， βm),∑i=1m βi =∑i=1m αm， β的排序在优势阶上是严格大于λ的划分。v的三角矩阵表示可以进行详细的研究。在空间pα上存在一个内积结构和一个方便的自伴随算子集合tiρ i，其中ρipα是p（α1, ....）的对象， αi + 1，…， αm)。这种结构与m个变量的杰克多项式具有双正交关系。v不存在的k的值称为奇异值，由de Jeu， Opdam和dunkl在Trans中研究。美国人。数学。社会科学。346(1994),237-256。作为该论文猜想的部分验证，我们构造，对于anya=1,2,3，…使得gcd(N−m+1,a)<(N−m+1)/m且m≤N/2，这是一个多项式空间，它被每个ti叉= - a/(N−m+1)湮灭，对称群psn在其上按照表示（N−m, m）作用。{}
AbstractThere 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
$$T_i : = frac{partial }{{partial x_i }} + ksum
olimits_{j
e i} {frac{{1 - (ij)}}{{x_i - x_j }}} $$
, (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
$$T_i V = Vfrac{partial }{{partial x_i }}$$
,i = 1, ...,N, normalized byV1=1 (seeDunkl, Canadian J. Math.43 (1991), 1213–1227). We show thatTipα=0 fori>m, and
$$V(x_1^{alpha _1 } cdots x_m^{alpha _m } ) = frac{{lambda _1 !lambda _2 ! cdots lambda _m !}}{{left( {Nk + 1}
ight)_{lambda _1 } left( {Nk - k + 1}
ight)_{lambda _2 } cdots (Nk - (m - 1)k + 1)_{lambda _m } }}p_alpha + sumlimits_eta {A_{eta alpha } p_{eta ,} } $$
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).