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 $\displaystyle T_l$ (vector) transforms as

$\displaystyle \Gamma(T_l) = A_2 \oplus E$ (It must transfom as (x,y) and z).

A tensor of rank two then transform as

$\displaystyle \Gamma(T_{lm}) = \Gamma(T_lT_m) = (A_2 \oplus E) \otimes (A_2 \oplus E) = 2A1 \oplus A_2 \oplus 3E$ (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 $\displaystyle T_{lm}$ is symmetric (i.e. $\displaystyle T_{lm} = T_{ml}$) it is easyly seen that

$\displaystyle \Gamma([T_{lm}]^S) = 2A1 \oplus 2E$

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 $\displaystyle A_2$ and one $\displaystyle E$, and not for example one $\displaystyle A_1$ and $\displaystyle E$, or two $\displaystyle A_1$ and one $\displaystyle A_2$ 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