@inbook{10.2307/j.ctt7rnjq.13,
ISBN = {9780691118369},
URL = {http://www.jstor.org/stable/j.ctt7rnjq.13},
abstract = {Orthogonal groupSO(n) is the group of transformations that leaves invariant a symmetric quadratic form (q,q) =gμνqμqν:If (q,q) is an invariant, so is its complex conjugate (q,q)*=gμνqμqν, andis also an invariant tensor. The matrix$A_\mu ^\nu = g_{\mu \sigma } g^{\sigma \nu } $must be proportional to unity, as otherwise its characteristic equation would decompose the definingn-dimensional rep. A convenient normalization isAs the indices can be raised and lowered at will, nothing is gained by keeping the arrows. Our convention will be to perform all contractions with metric tensors with upper indices and omit the arrows and the open dots:All},
bookauthor = {Predrag Cvitanović},
booktitle = {Group Theory: Birdtracks, Lie's, and Exceptional Groups},
pages = {118--131},
publisher = {Princeton University Press},
title = {Orthogonal groups},
year = {2008}
}