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
For any partition , the zonal polynomial is defined by the properties
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)).
Therefore is a symmetric polynomial in the eigenvalues of .
For and ,