Intertwining operators and polynomials associated with the symmetric group
Let \(T_i{:=}{\partial \over \partial x_i}+k\sum _{j\neq i} {1-\sigma _{ij} \over {x_i-x_j}}\), \(i=1,2,\dots ,N\), where \(k\) is a fixed parameter and \(\sigma _{ij}\) is the transposition of \(x_i\) and \(x_j\). These operators generate an algebra which is used in studying analytic structures invariant under permutation of coordinates. Let \(\{ p_\alpha \}\) be a family of functions indexed by \(m\)-tuples of non-negative integers \(\alpha =(\alpha _1,\alpha _2,\dots ,\alpha _m)\), \(m\leq N\). Let \(V\) be a linear map (an intertwining operator) preserving the degree of homogeneity for which \(T_iV=V {\partial \over \partial x_i}\). The author shows that \(T_ip_\alpha =0\) for \(i>m\) and gives the expression for \(V(x_1^{\alpha _1} \cdots x_m^{\alpha _m})\). The operator has triangular matrix representation. There exists an inner product structure on the span of \(\{ p_\alpha \}\) and a convenient set of self-adjoint operators \(\{ T_i\rho _i\} \), where \(\rho _i p_\alpha = p_{(\alpha _1,\dots ,\alpha _i+1,\dots ,\alpha _m)}\). This structure has a bi-orthogonal relationship with the Jack polynomials in \(m\) variables. For any \(a=1,2,\dots \) such that \(\text{ gcd} (N-m+1,a)< (N-m+1)/m\) and \(m\leq N/2\), the author constructs a space of polynomials annihilated by each \(T_i\) for \(k=-a/(N-m+1)\) and on which the symmetric group \(S_N\) acts according to the irreducible representation given by \((N-m,m)\).
scientific article; zbMATH DE number 1236106
scientific article