Concept question about the dimensionality of a representation in group theory here: Look at 3.1(c) of problem set, from group theory application to the physics of condensed matter of M.S.Dresselhaus:
3.1(c) Given the point group T,verify that the equality \begin{align} \sum_j{l_j^2} =h \end{align}
What is the meaning of the two sets of characters given for the two-dimensional irreducible representation E?
It seems that E with 2 sets of characters is called two-dimensional irreducible representation. But when verifying the equality above, where lj should be the dimensionality of the IR Γj, and E is viewed as TWO representations and each has a dimension lj=1(referring to the character of group T and p40 of Dresselhaus's book),so that we have : \begin{align} \sum_j{l_j^2} =h \\ 1^2+1^2+1^2+3^2 = 12 \end{align} to finish the proof. Instead of: \begin{align} 1^2+2^2+3^2=14 \neq 12 \end{align}
So, my question is what is the meaning of a 2 dimension representation? Becoz it seems that there are two definitions of dimension here.
The character table of group T attached here, from Dresselhaus's book