Jack vertex operators and realization of Jack functions

Abstract

We give an iterative method to realize general Jack functions from Jack functions of rectangular shapes. We first show some cases of Stanley's conjecture on positivity of the Littlewood-Richardson coefficients, and then use this method to give a new realization of Jack functions. We also show in general that vectors of products of Jack vertex operators form a basis of symmetric functions. In particular this gives a new proof of linear independence for the rectangular and marked rectangular Jack vertex operators. Thirdly a generalized Frobenius formula for Jack functions was given and was used to give new evaluation of Dyson integrals and even powers of Vandermonde determinant.Comment: Expanded versio

Macdonald symmetric functions of rectangular shapes

“…What really makes the exact complement interesting is the following property on Macdonald functions. This property is known in [5] (see Remark 4.9 there). Recall that for a symmetric function F ∈ Λ F , the operator F * is defined by F * .u, v = u, F v .…”

“…Taking the limit q → 1 in Theorem 5.2, we get the vertex operator realization of Jack function Q ρ (β −1 ) of almost rectangular shapes and also the following lowering operator formula (5.5) for them. This specification covers those corresponding results in [3] and [5], where the arguments work for ρ = ((k + 1) t , k s ) with s restricted to 0 and 1. Moreover, one can see in the following (from the proof) that for Jack function of almost rectangular shapes, the lowering operator (5.5) and the hyperdeterminant formula (5.4) are equivalent.…”

“…Definition 6.5. [5] For a rectangular partition R = (k s ), and a partition λ satisfying λ 1 ≤ k and l(λ) ≤ s, we define the complement λ = R − ′ λ of λ in R by (6.11) λ i = k − λ s+1−i for i = 1, 2, . .…”

“…To explain it explicitly, we first give the following: Definition 6.6. [5] For a partition λ = (a n 1 1 a n 2 2 · · · a nr r ) with a 1 > a 2 · · · > a r > 0 and n 1 , n 2 , . .…”

Combinatorial representation theory of Lie algebras. Richard Stanley’s work and the way it was continued

“…In [17], Knop and Sahi prove that the Jack structure constants are in Z[α], and give a monomial formula for J In [2], by studying the Laplace-Beltrami operator, Cai and Jing give another monomial formula (with rational function coefficients) for J (α) λ (x), as well as a (non-positive) formula for the Jack structure constants. In another paper, Cai and Jing [3] prove a special case of Richard's conjecture by deriving the generalization of the Pieri rule in which the one-rowed partition is replaced by a rectangle (or a rectangle and another row). Based on this formula, they settle the long-standing problem of realizing any J (α) λ (x) using vertex operators.…”