Jack function

Definition

The Jack function J κ ( α ) ( x 1 , x 2 , … ) of integer partition κ , parameter α , and indefinitely many arguments x 1 , x 2 , … , can be recursively defined as follows:

where the summation is over all partitions μ such that the skew partition κ / μ is a horizontal strip, namely

where B κ μ ν ( i , j ) equals κ j ′ − i + α ( κ i − j + 1 ) if κ j ′ = μ j ′ and κ j ′ − i + 1 + α ( κ i − j ) otherwise. The expressions κ ′ and μ ′ refer to the conjugate partitions of κ and μ , respectively. The notation ( i , j ) ∈ κ means that the product is taken over all coordinates ( i , j ) of boxes in the Young diagram of the partition κ .

Combinatorial formula

In 1997, F. Knop and S. Sahi gave a purely combinatorial formula for the Jack polynomials J μ ( α ) in n variables:

The sum is taken over all admissible tableaux of shape λ , and d T ( α ) = ∏ s ∈ T  critical d λ ( α ) ( s ) with d λ ( α ) ( s ) = α ( a λ ( s ) + 1 ) + ( l λ ( s ) + 1 ) .

An admissible tableau of shape λ is a filling of the Young diagram λ with numbers 1,2,…,n such that for any box (i,j) in the tableau,

A box s = ( i , j ) ∈ λ is critical for the tableau T if j>1 and T ( i , j ) = T ( i , j − 1 ) .

This result can be seen as a special case of the more general combinatorial formula for Macdonald polynomials.

C normalization

The Jack functions form an orthogonal basis in a space of symmetric polynomials, with inner product: ⟨ f , g ⟩ = ∫ [ 0 , 2 π ] n f ( e i θ 1 , ⋯ , e i θ n ) g ( e i θ 1 , ⋯ , e i θ n ) ¯ ∏ 1 ≤ j < k ≤ n | e i θ j − e i θ k | 2 / α d θ 1 ⋯ d θ n

This orthogonality property is unaffected by normalization. The normalization defined above is typically referred to as the J normalization. The C normalization is defined as

where

For α = 2 , C κ ( 2 ) ( x 1 , x 2 , … , x n ) denoted often as just C κ ( x 1 , x 2 , … , x n ) is known as the Zonal polynomial.

P normalization

The P normalization is given by the identity J λ = H λ ′ P λ , where H λ ′ = ∏ s ∈ λ ( α a λ ( s ) + l λ ( s ) + 1 ) and a λ and l λ denotes the arm and leg length respectively. Therefore, for α = 1 , P λ is the usual Schur function.

Similar to Schur polynomials, P λ can be expressed as a sum over Young tableaux. However, one need to add an extra weight to each tableau that depends on the parameter α .

Thus, a formula for the Jack function P λ is given by

where the sum is taken over all tableaux of shape λ , and T ( s ) denotes the entry in box s of T.

The weight ψ T ( α ) can be defined in the following fashion: Each tableau T of shape λ can be interpreted as a sequence of partitions ∅ = ν 1 → ν 2 → ⋯ → ν n = λ where ν i + 1 / ν i defines the skew shape with content i in T. Then ψ T ( α ) = ∏ i ψ ν i + 1 / ν i ( α ) where

and the product is taken only over all boxes s in λ such that s has a box from λ / μ in the same row, but not in the same column.

Connection with the Schur polynomial

When α = 1 the Jack function is a scalar multiple of the Schur polynomial

where

is the product of all hook lengths of κ .

Properties

If the partition has more parts than the number of variables, then the Jack function is 0:

Matrix argument

In some texts, especially in random matrix theory, authors have found it more convenient to use a matrix argument in the Jack function. The connection is simple. If X is a matrix with eigenvalues x 1 , x 2 , … , x m , then