Digital Library of Mathematical Functions
About the Project
NIST
35 Functions of Matrix ArgumentProperties

§35.4 Partitions and Zonal Polynomials

Contents

§35.4(i) Definitions

A partition κ=(k1,,km) is a vector of nonnegative integers, listed in nonincreasing order. Also, |κ| denotes k1++km, the weight of κ; (κ) denotes the number of nonzero kj; a+κ denotes the vector (a+k1,,a+km).

The partitional shifted factorial is given by

35.4.1 [a]κ=Γm(a+κ)Γm(a)=j=1m(a-12(j-1))kj,

where (a)k=a(a+1)(a+k-1).

For any partition κ, the zonal polynomial Zκ:𝒮 is defined by the properties

35.4.2 Zκ(I)=|κ|! 22|κ|[m/2]κ1j<l(κ)(2kj-2kl-j+l)j=1(κ)(2kj+(κ)-j)!

and

35.4.3 Zκ(T)=Zκ(I)|T|kmO(m)j=1m-1|(HTH-1)j|kj-kj+1H,
T𝒮.

See Muirhead (1982, pp. 68–72) for the definition and properties of the Haar measure 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 Cκ(T) (Muirhead (1982, pp. 227–239)), 𝒴κ(T) (Takemura (1984, p. 22)), and Φκ(T) (Faraut and Korányi (1994, pp. 228–236)).

§35.4(ii) Properties

Normalization

35.4.4 Zκ(0)={1,κ=(0,,0),0,κ(0,,0).

Orthogonal Invariance

35.4.5 Zκ(HTH-1)=Zκ(T),
HO(m).

Therefore Zκ(T) is a symmetric polynomial in the eigenvalues of T.

Summation

For k=0,1,2,,

35.4.6 |κ|=kZκ(T)=(trT)k.

Mean-Value

35.4.7 O(m)Zκ(SHTH-1)H=Zκ(S)Zκ(T)Zκ(I).

Laplace and Beta Integrals

For TΩ and (a),(b)>12(m-1),

35.4.8 Ωetr(-TX)|X|a-12(m+1)Zκ(X)X=Γm(a+κ)|T|-aZκ(T-1),
35.4.9 0<X<I|X|a-12(m+1)|I-X|b-12(m+1)Zκ(TX)X=[a]κ[a+b]κBm(a,b)Zκ(T).