Orthogonal group*SO*(*n*) is the group of transformations that leaves invariant a symmetric quadratic form (*q*,*q*) =*g*_{μν}*q*^{μ}*q*^{ν}:

If (*q*,*q*) is an invariant, so is its complex conjugate (*q*,*q*)^{*}=*g*^{μν}*q*_{μ}*q*_{ν}, and

is also an invariant tensor. The matrix*n*-dimensional rep. A convenient normalization is

As 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

EP - 131 PB - Princeton University Press PY - 2008 SN - 9780691118369 SP - 118 T2 - Group Theory T3 - Birdtracks, Lie's, and Exceptional Groups UR - http://www.jstor.org/stable/j.ctt7rnjq.13 Y2 - 2020/09/26/ ER -