Jack function

In mathematics, the Jack function is a generalization of the Jack polynomial, introduced by Henry Jack. The Jack polynomial is a homogeneous, symmetric polynomial which generalizes the Schur and zonal polynomials, and is in turn generalized by the Heckman–Opdam polynomials and Macdonald polynomials.

The Jack function J κ ( α ) ( x 1 , x 2 , … , x m ) {\displaystyle J_{\kappa }^{(\alpha )}(x_{1},x_{2},\ldots ,x_{m})} of an integer partition κ {\displaystyle \kappa } , parameter α {\displaystyle \alpha } , and arguments x 1 , x 2 , … , x m {\displaystyle x_{1},x_{2},\ldots ,x_{m}} can be recursively defined as follows:

For m=1

J k ( α ) ( x 1 ) = x 1 k ( 1 + α ) ⋯ ( 1 + ( k − 1 ) α ) {\displaystyle J_{k}^{(\alpha )}(x_{1})=x_{1}^{k}(1+\alpha )\cdots (1+(k-1)\alpha )}

For m>1

J κ ( α ) ( x 1 , x 2 , … , x m ) = ∑ μ J μ ( α ) ( x 1 , x 2 , … , x m − 1 ) x m | κ / μ | β κ μ , {\displaystyle J_{\kappa }^{(\alpha )}(x_{1},x_{2},\ldots ,x_{m})=\sum _{\mu }J_{\mu }^{(\alpha )}(x_{1},x_{2},\ldots ,x_{m-1})x_{m}^{|\kappa /\mu |}\beta _{\kappa \mu },}

where the summation is over all partitions μ {\displaystyle \mu } such that the skew partition κ / μ {\displaystyle \kappa /\mu } is a horizontal strip, namely

κ 1 ≥ μ 1 ≥ κ 2 ≥ μ 2 ≥ ⋯ ≥ κ n − 1 ≥ μ n − 1 ≥ κ n {\displaystyle \kappa _{1}\geq \mu _{1}\geq \kappa _{2}\geq \mu _{2}\geq \cdots \geq \kappa _{n-1}\geq \mu _{n-1}\geq \kappa _{n}} ( μ n {\displaystyle \mu _{n}} must be zero or otherwise J μ ( x 1 , … , x n − 1 ) = 0 {\displaystyle J_{\mu }(x_{1},\ldots ,x_{n-1})=0} ) and

β κ μ = ∏ ( i , j ) ∈ κ B κ μ κ ( i , j ) ∏ ( i , j ) ∈ μ B κ μ μ ( i , j ) , {\displaystyle \beta _{\kappa \mu }={\frac {\prod _{(i,j)\in \kappa }B_{\kappa \mu }^{\kappa }(i,j)}{\prod _{(i,j)\in \mu }B_{\kappa \mu }^{\mu }(i,j)}},}

where B κ μ ν ( i , j ) {\displaystyle B_{\kappa \mu }^{\nu }(i,j)} equals κ j ′ − i + α ( κ i − j + 1 ) {\displaystyle \kappa _{j}'-i+\alpha (\kappa _{i}-j+1)} if κ j ′ = μ j ′ {\displaystyle \kappa _{j}'=\mu _{j}'} and κ j ′ − i + 1 + α ( κ i − j ) {\displaystyle \kappa _{j}'-i+1+\alpha (\kappa _{i}-j)} otherwise. The expressions κ ′ {\displaystyle \kappa '} and μ ′ {\displaystyle \mu '} refer to the conjugate partitions of κ {\displaystyle \kappa } and μ {\displaystyle \mu } , respectively. The notation ( i , j ) ∈ κ {\displaystyle (i,j)\in \kappa } means that the product is taken over all coordinates ( i , j ) {\displaystyle (i,j)} of boxes in the Young diagram of the partition κ {\displaystyle \kappa } .

In 1997, F. Knop and S. Sahi ^[1] gave a purely combinatorial formula for the Jack polynomials J μ ( α ) {\displaystyle J_{\mu }^{(\alpha )}} in n variables:

J μ ( α ) = ∑ T d T ( α ) ∏ s ∈ T x T ( s ) . {\displaystyle J_{\mu }^{(\alpha )}=\sum _{T}d_{T}(\alpha )\prod _{s\in T}x_{T(s)}.}

The sum is taken over all admissible tableaux of shape λ , {\displaystyle \lambda ,} and

d T ( α ) = ∏ s ∈ T  critical d λ ( α ) ( s ) {\displaystyle d_{T}(\alpha )=\prod _{s\in T{\text{ critical}}}d_{\lambda }(\alpha )(s)}

with

d λ ( α ) ( s ) = α ( a λ ( s ) + 1 ) + ( l λ ( s ) + 1 ) . {\displaystyle d_{\lambda }(\alpha )(s)=\alpha (a_{\lambda }(s)+1)+(l_{\lambda }(s)+1).}

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

• T ( i , j ) ≠ T ( i ′ , j ) {\displaystyle T(i,j)\neq T(i',j)} whenever i ′ > i . {\displaystyle i'>i.}
• T ( i , j ) ≠ T ( i , j − 1 ) {\displaystyle T(i,j)\neq T(i,j-1)} whenever j > 1 {\displaystyle j>1} and i ′ < i . {\displaystyle i'<i.}

A box s = ( i , j ) ∈ λ {\displaystyle s=(i,j)\in \lambda } is critical for the tableau T if j > 1 {\displaystyle j>1} and T ( i , j ) = T ( i , j − 1 ) . {\displaystyle 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.

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 {\displaystyle \langle f,g\rangle =\int _{[0,2\pi ]^{n}}f\left(e^{i\theta _{1}},\ldots ,e^{i\theta _{n}}\right){\overline {g\left(e^{i\theta _{1}},\ldots ,e^{i\theta _{n}}\right)}}\prod _{1\leq j<k\leq n}\left|e^{i\theta _{j}}-e^{i\theta _{k}}\right|^{\frac {2}{\alpha }}d\theta _{1}\cdots d\theta _{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

C κ ( α ) ( x 1 , … , x n ) = α | κ | ( | κ | ) ! j κ J κ ( α ) ( x 1 , … , x n ) , {\displaystyle C_{\kappa }^{(\alpha )}(x_{1},\ldots ,x_{n})={\frac {\alpha ^{|\kappa |}(|\kappa |)!}{j_{\kappa }}}J_{\kappa }^{(\alpha )}(x_{1},\ldots ,x_{n}),}

where

j κ = ∏ ( i , j ) ∈ κ ( κ j ′ − i + α ( κ i − j + 1 ) ) ( κ j ′ − i + 1 + α ( κ i − j ) ) . {\displaystyle j_{\kappa }=\prod _{(i,j)\in \kappa }\left(\kappa _{j}'-i+\alpha \left(\kappa _{i}-j+1\right)\right)\left(\kappa _{j}'-i+1+\alpha \left(\kappa _{i}-j\right)\right).}

For α = 2 , C κ ( 2 ) ( x 1 , … , x n ) {\displaystyle \alpha =2,C_{\kappa }^{(2)}(x_{1},\ldots ,x_{n})} is often denoted by C κ ( x 1 , … , x n ) {\displaystyle C_{\kappa }(x_{1},\ldots ,x_{n})} and called the Zonal polynomial.

The P normalization is given by the identity J λ = H λ ′ P λ {\displaystyle J_{\lambda }=H'_{\lambda }P_{\lambda }} , where

H λ ′ = ∏ s ∈ λ ( α a λ ( s ) + l λ ( s ) + 1 ) {\displaystyle H'_{\lambda }=\prod _{s\in \lambda }(\alpha a_{\lambda }(s)+l_{\lambda }(s)+1)}

where a λ {\displaystyle a_{\lambda }} and l λ {\displaystyle l_{\lambda }} denotes the arm and leg length respectively. Therefore, for α = 1 , P λ {\displaystyle \alpha =1,P_{\lambda }} is the usual Schur function.

Similar to Schur polynomials, P λ {\displaystyle P_{\lambda }} 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 {\displaystyle \alpha } .

Thus, a formula ^[2] for the Jack function P λ {\displaystyle P_{\lambda }} is given by

P λ = ∑ T ψ T ( α ) ∏ s ∈ λ x T ( s ) {\displaystyle P_{\lambda }=\sum _{T}\psi _{T}(\alpha )\prod _{s\in \lambda }x_{T(s)}}

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

The weight ψ T ( α ) {\displaystyle \psi _{T}(\alpha )} can be defined in the following fashion: Each tableau T of shape λ {\displaystyle \lambda } can be interpreted as a sequence of partitions

∅ = ν 1 → ν 2 → ⋯ → ν n = λ {\displaystyle \emptyset =\nu _{1}\to \nu _{2}\to \dots \to \nu _{n}=\lambda }

where ν i + 1 / ν i {\displaystyle \nu _{i+1}/\nu _{i}} defines the skew shape with content i in T. Then

ψ T ( α ) = ∏ i ψ ν i + 1 / ν i ( α ) {\displaystyle \psi _{T}(\alpha )=\prod _{i}\psi _{\nu _{i+1}/\nu _{i}}(\alpha )}

where

ψ λ / μ ( α ) = ∏ s ∈ R λ / μ − C λ / μ ( α a μ ( s ) + l μ ( s ) + 1 ) ( α a μ ( s ) + l μ ( s ) + α ) ( α a λ ( s ) + l λ ( s ) + α ) ( α a λ ( s ) + l λ ( s ) + 1 ) {\displaystyle \psi _{\lambda /\mu }(\alpha )=\prod _{s\in R_{\lambda /\mu }-C_{\lambda /\mu }}{\frac {(\alpha a_{\mu }(s)+l_{\mu }(s)+1)}{(\alpha a_{\mu }(s)+l_{\mu }(s)+\alpha )}}{\frac {(\alpha a_{\lambda }(s)+l_{\lambda }(s)+\alpha )}{(\alpha a_{\lambda }(s)+l_{\lambda }(s)+1)}}}

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

When α = 1 {\displaystyle \alpha =1} the Jack function is a scalar multiple of the Schur polynomial

J κ ( 1 ) ( x 1 , x 2 , … , x n ) = H κ s κ ( x 1 , x 2 , … , x n ) , {\displaystyle J_{\kappa }^{(1)}(x_{1},x_{2},\ldots ,x_{n})=H_{\kappa }s_{\kappa }(x_{1},x_{2},\ldots ,x_{n}),}

where

H κ = ∏ ( i , j ) ∈ κ h κ ( i , j ) = ∏ ( i , j ) ∈ κ ( κ i + κ j ′ − i − j + 1 ) {\displaystyle H_{\kappa }=\prod _{(i,j)\in \kappa }h_{\kappa }(i,j)=\prod _{(i,j)\in \kappa }(\kappa _{i}+\kappa _{j}'-i-j+1)}

is the product of all hook lengths of κ {\displaystyle \kappa } .

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

J κ ( α ) ( x 1 , x 2 , … , x m ) = 0 ,  if  κ m + 1 > 0. {\displaystyle J_{\kappa }^{(\alpha )}(x_{1},x_{2},\ldots ,x_{m})=0,{\mbox{ if }}\kappa _{m+1}>0.}

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 {\displaystyle X} is a matrix with eigenvalues x 1 , x 2 , … , x m {\displaystyle x_{1},x_{2},\ldots ,x_{m}} , then

J κ ( α ) ( X ) = J κ ( α ) ( x 1 , x 2 , … , x m ) . {\displaystyle J_{\kappa }^{(\alpha )}(X)=J_{\kappa }^{(\alpha )}(x_{1},x_{2},\ldots ,x_{m}).}

• Demmel, James; Koev, Plamen (2006), "Accurate and efficient evaluation of Schur and Jack functions", Mathematics of Computation, 75 (253): 223–239, CiteSeerX 10.1.1.134.5248, doi:10.1090/S0025-5718-05-01780-1, MR 2176397.
• Jack, Henry (1970–1971), "A class of symmetric polynomials with a parameter", Proceedings of the Royal Society of Edinburgh, Section A. Mathematics, 69: 1–18, MR 0289462.
• Knop, Friedrich; Sahi, Siddhartha (19 March 1997), "A recursion and a combinatorial formula for Jack polynomials", Inventiones Mathematicae, 128 (1): 9–22, arXiv:q-alg/9610016, Bibcode:1997InMat.128....9K, doi:10.1007/s002220050134, S2CID 7188322
• Macdonald, I. G. (1995), Symmetric functions and Hall polynomials, Oxford Mathematical Monographs (2nd ed.), New York: Oxford University Press, ISBN 978-0-19-853489-1, MR 1354144
• Stanley, Richard P. (1989), "Some combinatorial properties of Jack symmetric functions", Advances in Mathematics, 77 (1): 76–115, doi:10.1016/0001-8708(89)90015-7, MR 1014073.

• Software for computing the Jack function by Plamen Koev and Alan Edelman.
• MOPS: Multivariate Orthogonal Polynomials (symbolically) (Maple Package) Archived 2010-06-20 at the Wayback Machine
• SAGE documentation for Jack Symmetric Functions