Question

In symmetry and group theory, why some representations are written as Ag, Au, Bg, Bu ,although some are A1, A2, B1, B2. I can't figure out the exact difference of them.

Answer 1

Lets illustrate this with help of two very familliar example.

The symbol A and B is decided by looking at the character of the principal axis rotation operation.

For example, In point group, the symmetry operations are

the representation that has character +1 in has the symbol A, thus those that are symmetric about rotation through principal axis are written as A. Those that are antisymmetric about the principal rotation axis (C_2 in case of C_2h group) are written B and they have character -1.

If we look at the character table for .

			i
	+1	+1	+1	+1
	+1	+1	-1	-1
	+1	-1	+1	-1
	+1	-1	-1	+1

In the above, the representations corresponds to representations with character +1 for the class i(inversion center). In other words, the representations that are symmetric with respect to center of inversion are written with a subscript 'g'.

Similarly, the representations that are anti-symmetric with respect to center of inversion(character -1) are written with a subscript 'u'.

.

Now, lets have a look at the character table

	+1	+1	+1	+1
	+1	+1	-1	-1
	+1	-1	+1	-1
	+1	-1	-1	+1

The Subscript 1 is added to representations that are symmetric about the plane, and have +1 character for that operation. Thus, the representation is symmetric about the plane of reflection in xz. A1 and B1 representations are symmetric about the xz plane reflection.

Similarly, those that are anti-symmetric abut the plane of reflection xz are written with a subscript 2. B2 and A2 representations are anti-symmetric about reflection about xz plane.