§35.4 Partitions and Zonal Polynomials

Keywords:: zonal polynomials
Permalink:: http://dlmf.nist.gov/35.4

§35.4(i) Definitions
§35.4(ii) Properties

§35.4(i) Definitions

Defines:: $\mathop{Z_{{\kappa}}\/}\nolimits\!\left(\mathbf{T}\right)$ : zonal polynomial
Keywords:: Haar measure, partitional shifted factorial, partitions, zonal polynomials
Referenced by:: §35.1
Permalink:: http://dlmf.nist.gov/35.4.i

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:: $\left[a\right]_{{\kappa}}$ : partitional shifted factorial
Symbols:: $\left(a\right)_{{n}}$ : Pochhammer’s symbol, $\mathop{\Gamma _{{m}}\/}\nolimits\!\left(a\right)$ : 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

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

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

35.4.2 $\mathop{Z_{{\kappa}}\/}\nolimits\!\left(\mathbf{I}\right)=|\kappa|!\, 2^{{2|\kappa|}}\,\left[m/2\right]_{{\kappa}}\frac{\prod\limits _{{1\leq j<l\leq\ell(\kappa)}}(2k_{j}-2k_{l}-j+l)}{\prod\limits _{{j=1}}^{{\ell(\kappa)}}(2k_{j}+\ell(\kappa)-j)!}$

Symbols:: $!$ : $n!$ : factorial, $\left[a\right]_{{\kappa}}$ : partitional shifted factorial, $\mathop{Z_{{\kappa}}\/}\nolimits\!\left(\mathbf{T}\right)$ : zonal polynomial, $j$ : nonnegative integer, $k$ : nonnegative integer, $m$ : positive integer and $\ell(\kappa)$ : number of nonzeros
Referenced by:: §35.4(i)
Permalink:: http://dlmf.nist.gov/35.4.E2
Encodings:: TeX, pMML, png

and

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

Symbols:: $dx$ : differential of $x$ , $\in$ : element of, $\int$ : integral, $\mathop{Z_{{\kappa}}\/}\nolimits\!\left(\mathbf{T}\right)$ : zonal polynomial, $\mathbf{O}(m)$ : space of orthogonal matrices, $j$ : nonnegative integer, $k$ : nonnegative integer, $m$ : positive integer and $\boldsymbol{\mathcal{S}}$ : space of real symmetric matrices
Referenced by:: §35.4(i)
Permalink:: http://dlmf.nist.gov/35.4.E3
Encodings:: TeX, pMML, png

See Muirhead (1982, pp. 68–72) for the definition and properties of the Haar measure $d\mathbf{H}$ . 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

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

¶ Normalization

Keywords:: zonal polynomials

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:: $\mathop{Z_{{\kappa}}\/}\nolimits\!\left(\mathbf{T}\right)$ : zonal polynomial
Permalink:: http://dlmf.nist.gov/35.4.E4
Encodings:: TeX, pMML, png

¶ Orthogonal Invariance

Keywords:: zonal polynomials

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

Symbols:: $\in$ : element of, $\mathop{Z_{{\kappa}}\/}\nolimits\!\left(\mathbf{T}\right)$ : zonal polynomial, $\mathbf{O}(m)$ : space of orthogonal matrices and $m$ : positive integer
Permalink:: http://dlmf.nist.gov/35.4.E5
Encodings:: TeX, pMML, png

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

¶ Summation

Keywords:: zonal polynomials

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

35.4.6 $\sum _{{|\kappa|=k}}\mathop{Z_{{\kappa}}\/}\nolimits\!\left(\mathbf{T}\right)=(\trace{\mathbf{T}})^{k}.$

Symbols:: $\trace\mathbf{A}$ : trace of matrix $\mathbf{A}$ , $\mathop{Z_{{\kappa}}\/}\nolimits\!\left(\mathbf{T}\right)$ : zonal polynomial and $k$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/35.4.E6
Encodings:: TeX, pMML, png

¶ Mean-Value

Keywords:: zonal polynomials

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

Symbols:: $dx$ : differential of $x$ , $\int$ : integral, $\mathop{Z_{{\kappa}}\/}\nolimits\!\left(\mathbf{T}\right)$ : zonal polynomial, $\mathbf{O}(m)$ : space of orthogonal matrices and $m$ : positive integer
Referenced by:: §35.9
Permalink:: http://dlmf.nist.gov/35.4.E7
Encodings:: TeX, pMML, png

¶ Laplace and Beta Integrals

Keywords:: zonal polynomials

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

35.4.8 $\int _{{\boldsymbol{\Omega}}}\mathop{\mathrm{etr}\/}\nolimits\!\left(-\mathbf{T}\mathbf{X}\right)\,|\mathbf{X}|^{{a-\frac{1}{2}(m+1)}}\mathop{Z_{{\kappa}}\/}\nolimits\!\left(\mathbf{X}\right)d\mathbf{X}=\mathop{\Gamma _{{m}}\/}\nolimits\!\left(a+\kappa\right)\,|\mathbf{T}|^{{-a}}\mathop{Z_{{\kappa}}\/}\nolimits\!\left(\mathbf{T}^{{-1}}\right),$

Symbols:: $dx$ : differential of $x$ , $\mathop{\mathrm{etr}\/}\nolimits\!\left(\mathbf{X}\right)$ : exponential of trace, $\int$ : integral, $\mathop{\Gamma _{{m}}\/}\nolimits\!\left(a\right)$ : multivariate gamma function, $\mathop{Z_{{\kappa}}\/}\nolimits\!\left(\mathbf{T}\right)$ : zonal polynomial, $a$ : complex variable, ${\boldsymbol{\Omega}}$ : space of matrices and $m$ : positive integer
Permalink:: http://dlmf.nist.gov/35.4.E8
Encodings:: TeX, pMML, png

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

Symbols:: $dx$ : differential of $x$ , $\int$ : integral, $\mathop{\mathrm{B}_{{m}}\/}\nolimits\!\left(a,b\right)$ : multivariate beta function, $\left[a\right]_{{\kappa}}$ : partitional shifted factorial, $\mathop{Z_{{\kappa}}\/}\nolimits\!\left(\mathbf{T}\right)$ : zonal polynomial, $a$ : complex variable, $b$ : complex variable and $m$ : positive integer
Permalink:: http://dlmf.nist.gov/35.4.E9
Encodings:: TeX, pMML, png