The definition is given below:
If the definition of the Spaces of Matrix Elements is as given below:
But I do not understand why:
1- Any linear combination of matrix elements can be expressed invariantly (without using coordinates) as given in Eq.(3), could anyone explain this for me please?
2-Also I do not understand the paragraph under eq.(3), and why it is c_{ji} and not c_{ij}, could anyone explain this for me with a concrete example please?
3-Could anyone give me a concrete example describing the difference between the space of matrix elements of the representation $T$ and the matrix elements of $T$?
Thank you!
Best Answer
The point is that the operation "trace" is independent on the coordinates. That is if $A$ is a matrix and $B$ is obtained from $A$ by changing basis then $B=PAP^{-1}$ and $\text{trace}(A)=\text{trace}(B)$.
It's minor and technical, if you don't take the transposed one then you would get $\sum_{i,j} c_{j,i}T_{i,j}$ instead. Let me explain (I'm going to use an unusual letters for the indices in the beginning so you will see the correct picture in the end): If you take $C=[c_{j,l}]$ and $T(g)=[T_{j,l}(g)]$ then $$CT(g)_{j,l} = \sum_{i=1}^n c_{j,i}T_{i,l}(g)$$ and so the trace would be $$\sum_{j=1}^n CT(g)_{j,j} = \sum_{j=1}^n \sum_{i=1}^n c_{j,i}T_{i,j}(g)$$
But as you can see the indices are not the way the author wanted them to be, they're $c_{j,i}T_{j,i}$ instead of $c_{j,i}T_{j,i}$. In order to overcome this minor issue the author decided to multiply by the transposed matrix.
Before I answer this question let me explain:
Given any $T:G\rightarrow GL(V)$ the matrix coefficients are functions from $G$ to $\mathbb{C}$. The space of matrix elements are not just matrix coefficients, but also their linear combinations.
Consider the trivial example: Take $G$ to be any group, and $T:G\rightarrow GL(V)$ be trivial. In that case no matter which base you choose, there is only one matrix coordinate which is $\varphi(g)=n$ (A constant map from $G$ to $\mathbb{C}$, where $n$ is the dimension of $V$ which is also the trace of the identity matrix). However the space of matrix elements contains any linear combination as well, in particular it contains $2\varphi(g)=2n$ (which is another constant map with different constant).