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Maximal Groups on Which the Permanent is Multiplicative

Published online by Cambridge University Press:  20 November 2018

Leroy B. Beasley
Leroy B. Beasley*
Affiliation:
University of British Columbia, Vancouver, B.C.
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Let Δn be the set of all n × n, non-singular matrices of the form PD, where P is a permutation matrix and D is a diagonal matrix with complex entries. In (1, conjecture 12), Marcus and Mine asked: Is Δn a maximal group on which the permanent function is multiplicative? (that is, per AB = per A per B). The field over which the entries range was not mentioned in the conjecture; however, we assume that the complex number field was intended. Corollary 1 answers this in the affirmative. In fact, Δn is the only maximal group (or semigroup) on which the permanent is multiplicative. Let ρi be the set of all non-zero entries in the ith row and let λj be the set of all non-zero entries in the jth column.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 21 , 1969 , pp. 495 - 497
Copyright
Copyright © Canadian Mathematical Society 1969

References

1. Marcus, M. and Mine, H., Permanents, Amer. Math. Monthly 72 (1965), 577–591.Google Scholar