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On induced permutation matrices and the symmetric group

Published online by Cambridge University Press:  20 January 2009

A. C. Aitken
A. C. Aitken
Affiliation:
University of Edinburgh.
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The n! operations Ai of permutations upon n different ordered symbols correspond to n! matrices Ai of the nth order, which have in each row and in each column only one non-zero element, namely a unit. Such matrices Ai are called permutation matrices, since their effect in premultiplying an arbitrary column vector x = {x1x2….xn} is to impress the permutation Ai upon the elements xi. For example the six matrices of the third order

are permutation matrices. It is convenient to denote them by

where the bracketed indices refer to the permutations of natural order. Clearly the relation Ai Aj = Ak entails the matrix relation AiAj = Ak; in other words, the n! matrices Ai, give a matrix representation of the symmetric group of order n!.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 5 , Issue 1 , October 1936 , pp. 1 - 13
Copyright
Copyright © Edinburgh Mathematical Society 1936

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