Hello I'm reading a book about symmetry of solids in physics. In this book they explain that a tensor can be decomposed into the representations in the following way.

Lets look at the D3 point group (d3 - Point Group Symmetry Character Tables - Chemistry Online Education). From the character table it is easily seen that a one rank tensor (vector) transforms as

(It must transfom as (x,y) and z).

A tensor of rank two then transform as

(Here the product table for D3 has been used, see link)

So this I think I understand, my problem is that they then state that if is symmetric (i.e. ) it is easyly seen that

I fail to see this. I can count the number of independent components a in get 2*1+2*2 = 6, which makes sence, because if one think of the tensor as a matrix, a symmetric matrix exactly has 6 independent components, however i fail to see why it is I have to remove and one , and not for example one and , or two and one which both would result in 6 independetn components.

Could anyone please elaborate on this, it would be much appreciated.

ps. hope this is in the right section of the forum.

Anders Berthelsen