Evolution on sequence space and tensor products of representation spaces
This paper is not intended to prove new theorems, but to point out a surprising connection between evolutionary dynamics on sequence space as perceived by \textit{M. Eigen} [cf. Die Naturwissenschaften 58, 465-523 (1971)] and tensor products of representation spaces. The decisive observation is that the so-called Eigen evolution matrices which describe evolutionary dynamics on sequence space in terms of mutation and selection in an idealized form, can always be written as a product of a diagonal matrix and a matrix which is a \(\nu\)-fold Kronecker or tensor product of much smaller matrices. Here \(\nu\) is the fixed length of the sequences to be considered. This observation goes back to the second author and has already been extensively treated from the physico-chemical point of view in his paper, J. Math. Biol. 24, 667-680 (1987; Zbl 0605.92008)]. In this work we want to give the mathematical framework in which one can understand this observation and its applications most easily. As a by-product, we derive some interesting results concerning representations of the hyperoctahedral groups and similar wreath product groups, which should be folklore but do not seem to be easily available in the literature. The paper is organized as follows: In Section 2 we introduce the evolution problem; in Section 3 we recall some basic facts concerning the spectral decomposition of tensor products and apply these facts to Eigen evolution matrices; and in Section 4 we discuss the closely related represenation theoretic applications.
tensor products of representation spaces
representations of the hyperoctahedral groups
spectral decomposition of tensor products
scientific article; zbMATH DE number 4047486