Vertex Operator Realization Of Jack Functions

We define the vertex operator associated with Jack functions in the vertex operator algebra of rank one lattice. We call this operator Jack vertex operator. We derive a general formula for products of vertex operators in terms of generalized homogeneous symmetric functions. We derive a Jacobi-Trudi like formula for the Jack vertex operator, thus show that under certain conditions the set of vertex operator products is indeed a basis for the representation space. As an application we reprove Mimachi-Yamada's theorem about the realization of Jack symmetric functions of rectangular shapes, and then generalized it to the shapes of a rectangular with the square in the lower right corner moved off. Furthermore, we realized the Jack functions for the partitions of rectangular shape with part of the last row removed. At the special case ofα=1 our new vertex operator provide another realization of the Schur Functions of the shape of a rectangular with part of last column removed. Acting on a proper term in the group algebra, the vertex operator products are symmetric functions with weight independent of a, we prove that these products are linearly independent thus they form a basis of the representation space over Q(a). Besides that some of the products are Jack functions, this basis has some other nice properties.We construct an infinite-variables version of Laplace-Beltrami type operator for Jack function. We prove that Jack functions are eigenvectors of this operator. Then we give a description of Jack symmetric functions with respect to this operator. We derive an explicit formula for the action of this operator on the monomial symmetric functions and on the generalized homogeneous symmetric functions. Basing on this operator, we find an iterating formula for the coefficient of monomials in the enpension of Jack symmetric func-tions. Then we find a combinatorial formula for Jack symmetric functions. We formulate a determinantal expression for Jack symmetric functions involving the generalized homo-geneous symmetric functions. Combinatorially, we formulate the Littlewood-Richardson coefficient of Jack symmetric functions. As an application, we regain the explicit formula for the Jack functions of shape two rows and those of two columns, we express them as the linear combinations of generalized homogeneous symmetric functions and the mono-mials respectively. On the representation of Virasoro algebra on the Fock space, we find an explicit formula for the Virasoro algebra acting on Jack functions. As a by-product of this, we recover Mimachi-Yamada's results about the rectangular Jack functions being the unique(up to scalar) singular vector of Virasoro algebra.We find a method to compute the coefficient of the monomials of the even power of Vandermonde determinant with the tool of the vertex operator realization of rectangular Jack symmetric functions and the tool of the operator having Jack functions as it's eigenfunctions. We list some examples the coefficients of the monomials in the even power of Vandermonde determinant, as a demonstration of our methods to attack the the problem of coefficients and in hope that a formula of the generalized form of the coefficients could be conjectured and proved at last.