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In my understanding, in group theory the one-dimensional irrep A differs from the one-dimensional irrep B in the symmetry under rotation about the principal axis: A is when the state is symmetric and B is when the state is antisymmetric under that rotation. However, I find in the character table of point group D that B1 has a character of 1 under the principal rotation operation while B2 and B3 have a character of -1.
Why the irrep is called B1 even though it has a character of 1 when rotated about the principal axis?
Why the irrep is called B1 even though it has a character of 1 when rotated about the principal axis?