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35 Functions of Matrix ArgumentProperties

§35.4 Partitions and Zonal Polynomials

  1. §35.4(i) Definitions
  2. §35.4(ii) Properties

§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(a12(j1))kj,

where (a)k=a(a+1)(a+k1).

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)(2kj2klj+l)j=1(κ)(2kj+(κ)j)!


35.4.3 Zκ(𝐓)=Zκ(𝐈)|𝐓|km𝐎(m)j=1m1|(𝐇𝐓𝐇1)j|kjkj+1d𝐇,

See Muirhead (1982, pp. 68–72) for the definition and properties of the Haar measure d𝐇. 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


35.4.4 Zκ(𝟎)={1,κ=(0,,0),0,κ(0,,0).

Orthogonal Invariance

35.4.5 Zκ(𝐇𝐓𝐇1)=Zκ(𝐓),

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


For k=0,1,2,,

35.4.6 |κ|=kZκ(𝐓)=(tr𝐓)k.


Laplace and Beta Integrals