I'm a physics student and I'm struggling with some concepts of Representation Theory.
I'm currently trying to understand the basics of it and in particular the mathematics that lies behind the concept of Reducible representation.
Let $G$ be a group. Let $X_i$, $i=1,\dots,n$ be vector spaces on the same field $\mathbb{K}$. For each $i$, let $D_i$ be a representation of $G$ on $X_i$. I have to show that the mapping $\oplus_iD_i\colon G\to \mathrm{End}(\oplus_iX_i)$ defined by
\begin{equation} (\oplus_iD_i)(g)=\oplus_iD_i(g), \qquad g\in G \end{equation} is a representation of $G$ in $\oplus_iX_i$.
To show that I need to recall that:
If $A_i\in \mathrm{End}(X_i)$ are endomorphisms then $\oplus _iA_i\in \mathrm{End}(\oplus _iX_i)$ and $(\oplus_iA_i)(\oplus_jx_j)=\oplus_i(A_ix_i)$
If $A_i\in \mathrm{End}(X_i)$ are invertible endomorphisms, then $\oplus_iA_i$ is invertible
If $A_i, B_i\in \mathrm{End}(X_i)$ then $(\oplus_i A_i )(\oplus_j B_j)=\oplus_i (A_i B_i)$.
I can't understand where these propositions come from. "Where" am I? I'm looking at a sum of objects that belongs to different monoids. Is it possible to generalize to every algebraic structure? For example, can I say something like
If $A_i, B_i\in X_i$ then $\oplus_iA_i\in\oplus_i X_i$ and $(\oplus_i A_i )+(\oplus_j B_j)=\oplus_i (A_i+ B_i)$
where "+" is the sum operation belonging to each vector space and $\oplus$ can freely move through the different spaces?
And lastly, what makes me say that $\oplus _iA_i\in \mathrm{End}(\oplus _iX_i)$ instead of $\oplus _iA_i\in \oplus _i\mathrm{End}(X_i)$?
I need some insight on the issue. As I said I'm not a math student so maybe I'm asking for the obvious... but any help, even if just a tip on where to take a look, will be greatly appreciated :)