§35.4 Partitions and Zonal Polynomials

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Keywords:: zonal polynomials
Permalink:: http://dlmf.nist.gov/35.4
See also:: Annotations for Ch.35

§35.4(i) Definitions

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Defines:: $Z_{\NVar{\kappa}}\left(\NVar{\mathbf{T}}\right)$ : zonal polynomial
Keywords:: Haar measure, definition, notation, partitional shifted factorial, partitions, weight of, zonal polynomials
Referenced by:: §35.1
Permalink:: http://dlmf.nist.gov/35.4.i
See also:: Annotations for §35.4 and Ch.35

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		$Z_{\kappa}\left(\mathbf{I}\right)=\|\kappa\|!\,2^{2\|\kappa\|}\,{\left[m/2\right]_% {\kappa}}\frac{\prod\limits_{1\leq j<l\leq\ell(\kappa)}(2k_{j}-2k_{l}-j+l)}{% \prod\limits_{j=1}^{\ell(\kappa)}(2k_{j}+\ell(\kappa)-j)!}$
ⓘ Symbols: $!$ : factorial (as in $n!$ ), ${\left[\NVar{a}\right]_{\NVar{\kappa}}}$ : partitional shifted factorial, $Z_{\NVar{\kappa}}\left(\NVar{\mathbf{T}}\right)$ : zonal polynomial, $\mathbf{I}$ : $m\times m$ identity matrix, $j$ : nonnegative integer, $k$ : nonnegative integer, $m$ : positive integer, $\kappa$ : vector of nonnegative integers and $\ell(\kappa)$ : number of nonzeros Referenced by: §35.4(i) Permalink: http://dlmf.nist.gov/35.4.E2 Encodings: TeX, pMML, png See also: Annotations for §35.4(i), §35.4 and Ch.35

and

35.4.3		$Z_{\kappa}\left(\mathbf{T}\right)=Z_{\kappa}\left(\mathbf{I}\right)\,\left\|% \mathbf{T}\right\|^{k_{m}}\*\int\limits_{\mathbf{O}(m)}\prod_{j=1}^{m-1}\|(% \mathbf{H}\mathbf{T}\mathbf{H}^{-1})_{j}\|^{k_{j}-k_{j+1}}\mathrm{d}{\mathbf{H}},$
$\mathbf{T}\in\boldsymbol{\mathcal{S}}$ .
ⓘ Symbols: $\in$ : element of, $\int$ : integral, $Z_{\NVar{\kappa}}\left(\NVar{\mathbf{T}}\right)$ : zonal polynomial, $\mathbf{I}$ : $m\times m$ identity matrix, $\boldsymbol{\mathcal{S}}$ : space of real symmetric $m\times m$ matrices, $\mathbf{T}$ : real symmetric $m\times m$ matrix, $\left\|\NVar{\mathbf{X}}\right\|$ : determinant of $\NVar{\mathbf{X}}$ , $\|(\NVar{\mathbf{X}})_{\NVar{j}}\|$ : $\NVar{j}$ th principal minor of $\NVar{\mathbf{X}}$ , $\mathbf{O}(m)$ : space of $m\times m$ orthogonal matrices, $\mathbf{H}$ : orthogonal $m\times m$ matrix, $\mathrm{d}{\mathbf{H}}$ : normalized Haar measure on $\mathbf{O}(m)$ , $j$ : nonnegative integer, $k$ : nonnegative integer, $m$ : positive integer and $\kappa$ : vector of nonnegative integers Referenced by: §35.4(i) Permalink: http://dlmf.nist.gov/35.4.E3 Encodings: TeX, pMML, png See also: Annotations for §35.4(i), §35.4 and Ch.35

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

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Notes:: See James (1964), Muirhead (1982, Chapter 7), Macdonald (1995, p. 425). See also Constantine (1963), Maass (1971, pp. 64–71), Macdonald (1995, pp. 388–439).
Permalink:: http://dlmf.nist.gov/35.4.ii
See also:: Annotations for §35.4 and Ch.35

Normalization

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Keywords:: normalization, zonal polynomials
See also:: Annotations for §35.4(ii), §35.4 and Ch.35

35.4.4		$Z_{\kappa}\left(\boldsymbol{{0}}\right)=\begin{cases}1,&\kappa=(0,\dots,0),\\ 0,&\kappa\neq(0,\dots,0).\end{cases}$
ⓘ Symbols: $Z_{\NVar{\kappa}}\left(\NVar{\mathbf{T}}\right)$ : zonal polynomial, $\boldsymbol{{0}}$ : $m\times m$ matrix of zeros and $\kappa$ : vector of nonnegative integers Permalink: http://dlmf.nist.gov/35.4.E4 Encodings: TeX, pMML, png See also: Annotations for §35.4(ii), §35.4(ii), §35.4 and Ch.35

Orthogonal Invariance

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Keywords:: orthogonality, zonal polynomials
See also:: Annotations for §35.4(ii), §35.4 and Ch.35

35.4.5		$Z_{\kappa}\left(\mathbf{H}\mathbf{T}\mathbf{H}^{-1}\right)=Z_{\kappa}\left(% \mathbf{T}\right),$
$\mathbf{H}\in\mathbf{O}(m)$ .
ⓘ Symbols: $\in$ : element of, $Z_{\NVar{\kappa}}\left(\NVar{\mathbf{T}}\right)$ : zonal polynomial, $\mathbf{T}$ : real symmetric $m\times m$ matrix, $\mathbf{O}(m)$ : space of $m\times m$ orthogonal matrices, $\mathbf{H}$ : orthogonal $m\times m$ matrix, $m$ : positive integer and $\kappa$ : vector of nonnegative integers Permalink: http://dlmf.nist.gov/35.4.E5 Encodings: TeX, pMML, png See also: Annotations for §35.4(ii), §35.4(ii), §35.4 and Ch.35

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

Summation

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Keywords:: summation, zonal polynomials
See also:: Annotations for §35.4(ii), §35.4 and Ch.35

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

35.4.6		$\sum_{\|\kappa\|=k}Z_{\kappa}\left(\mathbf{T}\right)=(\operatorname{tr}{\mathbf{% T}})^{k}.$
ⓘ Symbols: $\operatorname{tr}\NVar{\mathbf{A}}$ : trace of matrix, $Z_{\NVar{\kappa}}\left(\NVar{\mathbf{T}}\right)$ : zonal polynomial, $\mathbf{T}$ : real symmetric $m\times m$ matrix, $k$ : nonnegative integer and $\kappa$ : vector of nonnegative integers Permalink: http://dlmf.nist.gov/35.4.E6 Encodings: TeX, pMML, png See also: Annotations for §35.4(ii), §35.4(ii), §35.4 and Ch.35

Mean-Value

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Keywords:: mean-value, zonal polynomials
See also:: Annotations for §35.4(ii), §35.4 and Ch.35

35.4.7		$\int_{\mathbf{O}(m)}Z_{\kappa}\left(\mathbf{S}\mathbf{H}\mathbf{T}\mathbf{H}^{% -1}\right)\mathrm{d}{\mathbf{H}}=\frac{Z_{\kappa}\left(\mathbf{S}\right)Z_{% \kappa}\left(\mathbf{T}\right)}{Z_{\kappa}\left(\mathbf{I}\right)}.$
ⓘ Symbols: $\int$ : integral, $Z_{\NVar{\kappa}}\left(\NVar{\mathbf{T}}\right)$ : zonal polynomial, $\mathbf{I}$ : $m\times m$ identity matrix, $\mathbf{S}$ : real symmetric $m\times m$ matrix, $\mathbf{T}$ : real symmetric $m\times m$ matrix, $\mathbf{O}(m)$ : space of $m\times m$ orthogonal matrices, $\mathbf{H}$ : orthogonal $m\times m$ matrix, $\mathrm{d}{\mathbf{H}}$ : normalized Haar measure on $\mathbf{O}(m)$ , $m$ : positive integer and $\kappa$ : vector of nonnegative integers Referenced by: §35.9 Permalink: http://dlmf.nist.gov/35.4.E7 Encodings: TeX, pMML, png See also: Annotations for §35.4(ii), §35.4(ii), §35.4 and Ch.35

Laplace and Beta Integrals

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Keywords:: Laplace integral, beta integral, zonal polynomials
See also:: Annotations for §35.4(ii), §35.4 and Ch.35

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

35.4.8		$\int_{\boldsymbol{\Omega}}\operatorname{etr}\left(-\mathbf{T}\mathbf{X}\right)% \,\left\|\mathbf{X}\right\|^{a-\frac{1}{2}(m+1)}Z_{\kappa}\left(\mathbf{X}\right% )\,\mathrm{d}{\mathbf{X}}=\Gamma_{m}\left(a+\kappa\right)\,\left\|\mathbf{T}% \right\|^{-a}Z_{\kappa}\left(\mathbf{T}^{-1}\right),$
ⓘ Symbols: $\operatorname{etr}\left(\NVar{\mathbf{A}}\right)$ : exponential of trace, $\int$ : integral, $\Gamma_{\NVar{m}}\left(\NVar{a}\right)$ : multivariate gamma function, $Z_{\NVar{\kappa}}\left(\NVar{\mathbf{T}}\right)$ : zonal polynomial, $a$ : complex variable, $\mathbf{T}$ : real symmetric $m\times m$ matrix, $\mathbf{X}$ : real symmetric $m\times m$ matrix, $\left\|\NVar{\mathbf{X}}\right\|$ : determinant of $\NVar{\mathbf{X}}$ , $\,\mathrm{d}$ : differential of matrix, ${\boldsymbol{\Omega}}$ : space of positive-definite real symmetric $m\times m$ matrices, $m$ : positive integer and $\kappa$ : vector of nonnegative integers Permalink: http://dlmf.nist.gov/35.4.E8 Encodings: TeX, pMML, png See also: Annotations for §35.4(ii), §35.4(ii), §35.4 and Ch.35

35.4.9		$\int\limits_{\boldsymbol{{0}}<\mathbf{X}<\mathbf{I}}\left\|\mathbf{X}\right\|^{a% -\frac{1}{2}(m+1)}\*\left\|\mathbf{I}-\mathbf{X}\right\|^{b-\frac{1}{2}(m+1)}Z_{% \kappa}\left(\mathbf{T}\mathbf{X}\right)\,\mathrm{d}{\mathbf{X}}=\frac{{\left[% a\right]_{\kappa}}}{{\left[a+b\right]_{\kappa}}}\mathrm{B}_{m}\left(a,b\right)% Z_{\kappa}\left(\mathbf{T}\right).$
ⓘ Symbols: $\int$ : integral, $\mathrm{B}_{\NVar{m}}\left(\NVar{a},\NVar{b}\right)$ : multivariate beta function, ${\left[\NVar{a}\right]_{\NVar{\kappa}}}$ : partitional shifted factorial, $Z_{\NVar{\kappa}}\left(\NVar{\mathbf{T}}\right)$ : zonal polynomial, $a$ : complex variable, $\mathbf{I}$ : $m\times m$ identity matrix, $\mathbf{T}$ : real symmetric $m\times m$ matrix, $\mathbf{X}$ : real symmetric $m\times m$ matrix, $\left\|\NVar{\mathbf{X}}\right\|$ : determinant of $\NVar{\mathbf{X}}$ , $\,\mathrm{d}$ : differential of matrix, $b$ : complex variable, $<$ : less-than for matrices, $m$ : positive integer, $\boldsymbol{{0}}$ : $m\times m$ matrix of zeros and $\kappa$ : vector of nonnegative integers Permalink: http://dlmf.nist.gov/35.4.E9 Encodings: TeX, pMML, png See also: Annotations for §35.4(ii), §35.4(ii), §35.4 and Ch.35