Hostname: page-component-848d4c4894-4rdrl Total loading time: 0 Render date: 2024-07-08T01:09:30.318Z Has data issue: false hasContentIssue false

On induced permutation matrices and the symmetric group

Published online by Cambridge University Press:  20 January 2009

A. C. Aitken
Affiliation:
University of Edinburgh.
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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
Copyright
Copyright © Edinburgh Mathematical Society 1936

References

Franklin, F.Ainer. Jour. Math., 16 (1894) 205–6.CrossRefGoogle Scholar
Frobenius, G.Berlin Sitzungsb., (1900) 518–9.Google Scholar
Hurwitz, A.Math. Ann. 45 (1894) 381404 (391).CrossRefGoogle Scholar
Littlewood, D. E. and Richardson, A. R.Quart. J. of Math. (Oxford Series) 6 (1935) 185, 193.Google Scholar
Littlewood, D. E. and Richardson, A. R.Phil. Trans. Roy. Soc. (A) 233 (1934) 99141 (101113).Google Scholar
MacMahon, P. A.Combinatory Analysis (1916), Vol. ii, Chapter 1.Google Scholar
Muir, T.History of Determinants, Vol. 1, 341.Google Scholar
Schur, I.Berlin Dissertation (1901), Ueber eine Klasse von Matrizen, 4952 passim.Google Scholar
Young, A.Phil. Trans. Roy. Soc. (A) 234 (1935) 79114 (8086).Google Scholar