I had to look it up to be sure, but here's what I found:

Mulliken Symbols (from: Weisstein, Eric W. "Mulliken Symbols." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/MullikenSymbols.html)

Symbols used to identify irreducible representations of groups:

A = singly degenerate state which is symmetric with respect to rotation about the principal Cn axis,

B = singly degenerate state which is antisymmetric with respect to rotation about the principal Cn axis,

E = doubly degenerate,

T = triply degenerate,

Xg = (gerade, symmetric) the sign of the wavefunction does not change on inversion through the center of the atom,

Xu = (ungerade, antisymmetric) the sign of the wavefunction changes on inversion through the center of the atom,

X1 (on a or b) the sign of the wavefunction does not change upon rotation about the center of the atom,

X2 (on a or b) the sign of the wavefunction changes upon rotation about the center of the atom,

' = symmetric with respect to a horizontal symmetry plane ,

" = antisymmetric with respect to a horizontal symmetry plane .

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So, your '2' of 3A2g means there is a change in the sign of the wavefunction when you rotate about the centre of the atom

(just to check with the more common t2g and eg sets of an octahedral transition metal complex: The dxy, dyz and dxz (t2g) do undergo a change in wavefunction sign upon rotation, whereas dz2 and dx2-y2 do not)

may be it is because A orbitals are doubly degenerate.

I had to look it up to be sure, but here's what I found:

Mulliken Symbols (from: Weisstein, Eric W. "Mulliken Symbols." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/MullikenSymbols.html)

Symbols used to identify irreducible representations of groups:

A = singly degenerate state which is symmetric with respect to rotation about the principal Cn axis,

B = singly degenerate state which is antisymmetric with respect to rotation about the principal Cn axis,

E = doubly degenerate,

T = triply degenerate,

Xg = (gerade, symmetric) the sign of the wavefunction does not change on inversion through the center of the atom,

Xu = (ungerade, antisymmetric) the sign of the wavefunction changes on inversion through the center of the atom,

X1 (on a or b) the sign of the wavefunction does not change upon rotation about the center of the atom,

X2 (on a or b) the sign of the wavefunction changes upon rotation about the center of the atom,

' = symmetric with respect to a horizontal symmetry plane ,

" = antisymmetric with respect to a horizontal symmetry plane .

---

So, your '2' of 3A2g means there is a change in the sign of the wavefunction when you rotate about the centre of the atom

(just to check with the more common t2g and eg sets of an octahedral transition metal complex: The dxy, dyz and dxz (t2g) do undergo a change in wavefunction sign upon rotation, whereas dz2 and dx2-y2 do not)

Thanks, Ian for providing the answer. However, I have a small doubt. The sign for dz2 does not change upon rotation. But, I think the sign for dx2-y2 will change upon rotation. If you can clarify this issue, it would be very nice. Thanks, once again.

I also noticed this... (I'm just speculating but) it looks like the rotation associated with the 'subscript 2' is not necessarily around the principal axis, and it is important that the number is 2 rather than 3, or 4 etc:

dz2 when rotated 180 degrees about the x or y axes, or any angle about the z axis does not have any of its lobes change sign.

dx2-y2 when rotated by 180 degrees about the x, y or z axis again doesn't involve a change of sign. 

But, as you suggest, rotation of dx2-y2 about the z axis by 90 degrees does result in a change of sign. So, I think its probably important that the number is a '2', ie the rotation in question is 360/2 degrees ie 180 degrees, not 90 degrees. (This follows the same scheme as the the other numbers featured in point group notation e.g. C3 refers to a rotation by 360/3 degrees)

The subscripts are based on the perpendicular Cn axis. We are talking about a d8 sq.planar system (point group: D4h; with C4 as your major axis). If the rotation with respect to the perpendicular C2 is positive (as in dz2) then it is assigned 1; an anti-symmetric configuration would yield -1.

If this is a sq.planar system then the dx2-y2 will not be affected when you rotate it through the perpendicular C2 (denoted as C2'; passes through the bonds), which is why it has its Mulliken symbol as B1g. dxy, however, will be impacted by a perpendicular C2 and its Mulliken symbol is B2g.

Thanks. However, I should ask the question in more specific way. The ground state term symbol for [Ni(H2O)6 ]2+ is 3A2g. How does this 2 term come for t2g6 eg2 electronic configuration? If I remember correctly, it is due to a combination of (-)(-)(-)(-)(-)(-)(-)(+) and finally it is (-). (-) gives 2, where as (+) gives 1. For example, the ground state term symbol for [Mn(H2O)6]2+ is 6A1g. In this case, the combination is (-)(-)(-)(-)(+) due to t2g3 eg2 configuration and finally it is (+). However, I forgot to way to find out the sign concept. It should be related to rotation. But I am confused about the rotational axis. Whether it would be the principle or secondary axis?

7 orbitals of F are split into 3 energy levels when ligand interact with metal orbitals. See the F 5z3- 3zr2 orbital shape. Here principal axis is C4. If you rotate this orbital about nC2(which is perpendicular to principal axis) then lobes sign will be changed. That's why subscript 2 has been used.

We were discussing about the terms generating from d orbitals. Nickel belongs to d block element, not f block.

Please find the attachment

The wave function for F state have same symmetry as the wave function for the corresponding set of f orbitals. This is the reason why I mentioned splitting of F.