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By Nicholas Jackson

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Xστ (n) ) = P ( x1 , . . , x n ) P ( x σ (1) , . . , x σ ( n ) ) = P ( x σ (1) , . . , x σ ( n ) ) P ( x τ (1) , . . , x τ ( n ) ) P ( x1 , . . , x n ) P ( x1 , . . , x n ) = sign(σ) sign(τ ). The transposition (1 2) has negative sign, since sign(1 2) = P ( x2 , x1 , . . , x n ) x − x1 = 2 = −1. P ( x1 , x2 , . . 60 tells us that a transposition τ = ( a b) ∈ Sn can be written as a product (1 a)(1 b)(1 a), which also has sign −1. Therefore, by the multiplicativity condition above, if σ ∈ Sn can be decomposed as a product of an even number of transpositions, then sign(σ ) = +1, and if it decomposes as an odd number of transpositions, then sign(σ ) = −1.

The following two examples, of matrices with complex entries, have particular relevance to quantum mechanics. Wikimedia Commons / Ludwik Szacinski (1844–1894) After graduating in 1865 from the University of Christiania (now Oslo) with a general science degree, Marius Sophus Lie (1842–1899) dabbled in various subjects, including astronomy, zoology and botany, before settling on mathematics. In 1869, on the strength of his first published paper, Repräsentation der Imaginären der Plangeometrie, he won a scholarship to travel to Berlin and Paris, where he met and worked with several important mathematicians including Felix Klein (1849–1925) and Camille Jordan (1838–1922).

The latter is more compact and does allow us to see how individual objects are shuffled around by the permutation, and with a little bit of practice it becomes almost as easy to write down the composition of two permutations. 58 This sequence of transpositions will not, in general, be unique for a given permutation; even worse, two different such decompositions of a given permutation won’t necessarily even have the same number of transpositions. 51, page 24. 52, page 24. 53, page 26. 54, page 27.

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A course in abstract algebra by Nicholas Jackson


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