Evolution on sequence space and tensor products of representation spaces
type: Item
instance of: scholarly article
title: Evolution on sequence space and tensor products of representation spaces
zbMATH Open document ID: 0642.92009
DOI: 10.1007/BF00047282
author: Andreas W. M. Dress
author: David S. Rumschitzki
published in: Acta Applicandae Mathematicae
publication date: 1988-01-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 92D15
Mathematics Subject Classification ID: 20C30
Mathematics Subject Classification ID: 15A69
Mathematics Subject Classification ID: 15A18
Mathematics Subject Classification ID: 20C35
Mathematics Subject Classification ID: 92D25
zbMATH DE Number: 4047486
zbMATH Keywords: evolutionary dynamics on sequence space
zbMATH Keywords: tensor products of representation spaces
zbMATH Keywords: Eigen evolution matrices
zbMATH Keywords: mutation
zbMATH Keywords: selection
zbMATH Keywords: representations of the hyperoctahedral groups
zbMATH Keywords: wreath product groups
zbMATH Keywords: spectral decomposition of tensor products
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1007/bf00047282
OpenAlex ID: W2025334662
description: scientific article; zbMATH DE number 4047486