# §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 ${\left[a\right]_{\kappa}}=\frac{\Gamma_{m}\left(a+\kappa\right)}{\Gamma_{m}% \left(a\right)}=\prod_{j=1}^{m}{\left(a-\tfrac{1}{2}(j-1)\right)_{k_{j}}},$ ⓘ Defines: ${\left[\NVar{a}\right]_{\NVar{\kappa}}}$: partitional shifted factorial Symbols: ${\left(\NVar{a}\right)_{\NVar{n}}}$: Pochhammer’s symbol (or shifted factorial), $\Gamma_{\NVar{m}}\left(\NVar{a}\right)$: multivariate gamma function, $a$: complex variable, $j$: nonnegative integer, $k$: nonnegative integer, $m$: positive integer and $\kappa$: vector of nonnegative integers Permalink: http://dlmf.nist.gov/35.4.E1 Encodings: TeX, pMML, png See also: Annotations for §35.4(i), §35.4 and Ch.35

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

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

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

and

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

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 $Z_{\kappa}\left(\boldsymbol{{0}}\right)=\begin{cases}1,&\kappa=(0,\dots,0),\\ 0,&\kappa\neq(0,\dots,0).\end{cases}$

### Orthogonal Invariance

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

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

### Summation

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

 35.4.6 $\sum_{|\kappa|=k}Z_{\kappa}\left(\mathbf{T}\right)=(\operatorname{tr}{\mathbf{% T}})^{k}.$

### Mean-Value

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

### Laplace and Beta Integrals

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

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