§35.4 Partitions and Zonal Polynomials

§35.4(i) Definitions

A partition is a vector of nonnegative integers, listed in nonincreasing order. Also, denotes , the weight of ; denotes the number of nonzero ; denotes the vector .

The partitional shifted factorial is given by

where .

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

§35.4(ii) Properties

35.4.4

¶ Orthogonal Invariance

Therefore is a symmetric polynomial in the eigenvalues of .