# §35.4 Partitions and Zonal Polynomials

## §35.4(i) Definitions

A partition $\kappa=(k_{1},\dots,k_{m})$ is a vector of nonnegative integers, listed in nonincreasing order. Also, $|\kappa|$ denotes $k_{1}+\dots+k_{m}$, the weight of $\kappa$; $\ell(\kappa)$ denotes the number of nonzero $k_{j}$; $a+\kappa$ denotes the vector $(a+k_{1},\dots,a+k_{m})$.

The partitional shifted factorial is given by

 35.4.1 ${\left[a\right]_{\kappa}}=\frac{\mathop{\Gamma_{m}\/}\nolimits\!\left(a+\kappa% \right)}{\mathop{\Gamma_{m}\/}\nolimits\!\left(a\right)}=\prod_{j=1}^{m}{\left% (a-\tfrac{1}{2}(j-1)\right)_{k_{j}}},$ Defines: ${\left[\NVar{a}\right]_{\NVar{\kappa}}}$: partitional shifted factorial Symbols: ${\left(\NVar{a}\right)_{\NVar{n}}}$: Pochhammer’s symbol (or shifted factorial), $\mathop{\Gamma_{\NVar{m}}\/}\nolimits\!\left(\NVar{a}\right)$: multivariate gamma function, $a$: complex variable, $j$: nonnegative integer, $k$: nonnegative integer and $m$: positive integer Permalink: http://dlmf.nist.gov/35.4.E1 Encodings: TeX, pMML, png See also: Annotations for 35.4(i)

where ${\left(a\right)_{k}}=a(a+1)\cdots(a+k-1)$.

For any partition $\kappa$, the zonal polynomial $\mathop{Z_{\kappa}\/}\nolimits:\boldsymbol{\mathcal{S}}\to\mathbb{R}$ is defined by the properties

 35.4.2 $\mathop{Z_{\kappa}\/}\nolimits\!\left(\mathbf{I}\right)=|\kappa|!\,2^{2|\kappa% |}\,{\left[m/2\right]_{\kappa}}\frac{\prod\limits_{1\leq j

and

 35.4.3 $\mathop{Z_{\kappa}\/}\nolimits\!\left(\mathbf{T}\right)=\mathop{Z_{\kappa}\/}% \nolimits\!\left(\mathbf{I}\right)\,|\mathbf{T}|^{k_{m}}\*\int\limits_{\mathbf% {O}(m)}\prod_{j=1}^{m-1}|(\mathbf{H}\mathbf{T}\mathbf{H}^{-1})_{j}|^{k_{j}-k_{% j+1}}\mathrm{d}\mathbf{H},$ $\mathbf{T}\in\boldsymbol{\mathcal{S}}$.

See Muirhead (1982, pp. 68–72) for the definition and properties of the Haar measure $\mathrm{d}\mathbf{H}$. See Hua (1963, p. 30), Constantine (1963), James (1964), and Macdonald (1995, pp. 425–431) for further information on (35.4.2) and (35.4.3). Alternative notations for the zonal polynomials are $C_{\kappa}(\mathbf{T})$ (Muirhead (1982, pp. 227–239)), $\mathcal{Y}_{\kappa}(\mathbf{T})$ (Takemura (1984, p. 22)), and $\Phi_{\kappa}(\mathbf{T})$ (Faraut and Korányi (1994, pp. 228–236)).

## §35.4(ii) Properties

### Normalization

 35.4.4 $\mathop{Z_{\kappa}\/}\nolimits\!\left(\boldsymbol{{0}}\right)=\begin{cases}1,&% \kappa=(0,\dots,0),\\ 0,&\kappa\neq(0,\dots,0).\end{cases}$ Symbols: $\mathop{Z_{\NVar{\kappa}}\/}\nolimits\!\left(\NVar{\mathbf{T}}\right)$: zonal polynomial Permalink: http://dlmf.nist.gov/35.4.E4 Encodings: TeX, pMML, png See also: Annotations for 35.4(ii)

### Orthogonal Invariance

 35.4.5 $\mathop{Z_{\kappa}\/}\nolimits\!\left(\mathbf{H}\mathbf{T}\mathbf{H}^{-1}% \right)=\mathop{Z_{\kappa}\/}\nolimits\!\left(\mathbf{T}\right),$ $\mathbf{H}\in\mathbf{O}(m)$.

Therefore $\mathop{Z_{\kappa}\/}\nolimits\!\left(\mathbf{T}\right)$ is a symmetric polynomial in the eigenvalues of $\mathbf{T}$.

### Summation

For $k=0,1,2,\dots$,

 35.4.6 $\sum_{|\kappa|=k}\mathop{Z_{\kappa}\/}\nolimits\!\left(\mathbf{T}\right)=(% \mathop{\mathrm{tr}}{\mathbf{T}})^{k}.$

### Mean-Value

 35.4.7 $\int_{\mathbf{O}(m)}\mathop{Z_{\kappa}\/}\nolimits\!\left(\mathbf{S}\mathbf{H}% \mathbf{T}\mathbf{H}^{-1}\right)\mathrm{d}\mathbf{H}=\frac{\mathop{Z_{\kappa}% \/}\nolimits\!\left(\mathbf{S}\right)\mathop{Z_{\kappa}\/}\nolimits\!\left(% \mathbf{T}\right)}{\mathop{Z_{\kappa}\/}\nolimits\!\left(\mathbf{I}\right)}.$

### Laplace and Beta Integrals

For $\mathbf{T}\in{\boldsymbol{\Omega}}$ and $\Re{(a)},\Re{(b)}>\frac{1}{2}(m-1)$,

 35.4.8 $\int_{\boldsymbol{\Omega}}\mathop{\mathrm{etr}\/}\nolimits\!\left(-\mathbf{T}% \mathbf{X}\right)\,|\mathbf{X}|^{a-\frac{1}{2}(m+1)}\mathop{Z_{\kappa}\/}% \nolimits\!\left(\mathbf{X}\right)\mathrm{d}\mathbf{X}=\mathop{\Gamma_{m}\/}% \nolimits\!\left(a+\kappa\right)\,|\mathbf{T}|^{-a}\mathop{Z_{\kappa}\/}% \nolimits\!\left(\mathbf{T}^{-1}\right),$
 35.4.9 $\int\limits_{\boldsymbol{{0}}<\mathbf{X}<\mathbf{I}}|\mathbf{X}|^{a-\frac{1}{2% }(m+1)}\*|\mathbf{I}-\mathbf{X}|^{b-\frac{1}{2}(m+1)}\mathop{Z_{\kappa}\/}% \nolimits\!\left(\mathbf{T}\mathbf{X}\right)\mathrm{d}\mathbf{X}=\frac{{\left[% a\right]_{\kappa}}}{{\left[a+b\right]_{\kappa}}}\mathop{\mathrm{B}_{m}\/}% \nolimits\!\left(a,b\right)\mathop{Z_{\kappa}\/}\nolimits\!\left(\mathbf{T}% \right).$