Derivation of the eigenvalues of the configuration process induced by a labeled direct product branching process

Abstract

Ewens and Kirby in the previous paper determined the full set of eigenvalues in the (Wright) neutral alleles finite population model admitting an infinite number of possible allelic types Q where each gene mutation creates a novel type (see their paper for background discussion of the problem and other references). They use a method akin to that of identity by descent and some combinatorial devices. In this note the eigenvalues of the configuration process (defined precisely later) induced by a direct product branching process (which includes the Wright model) (Karlin and McGregor (1964); see also Karlin (1966, p. 416)) are derived from knowledge of the eigenstructure set forth in Karlin and McGregor (1966) on the labeled process, combined with an elementary symmetrization procedure. For completeness, we review the formulation and key facts needed, (see (3, pp. 396-416)). Consider a haploid population of N individuals composed from possible allelic types A, , A, ,..., A, . A state in the population process is described by a p-tuple, i = (ix , iZ ,..., i,) where i, is the number of occurrences of allele A, . Of course xF=, iV = N. The collection of all states is denoted by d. Suppose each individual of type A, in one generation produces progeny of all types according to the probability generating function