# further decomposing the canonical decomposition of a representation

I am trying to learn representation theory by myself,so please pardon me if this is dumb. let V be a representation of a finite group G, $W_1,...,W_h$ be the distinct irreducible representations of G, and let V = $U_1 \oplus ... \oplus U_m$ be some decomposition of V into irreducible subrepresentations. Then we can write V = $V_1\oplus ...\oplus V_h$, where $V_i$ is the direct sum of irreducible subrepresentations among $U_i$'s which are $isomorphic$ to $W_i$. Then it is shown that this last decomposition of V is unique.

I have understood this much, but I cannot understand what we can say about the decomposition of each $V_i$'s. The author says that the decomposition of each $V_i$ can be done in an infinitely many ways; it is just as arbitrary as choosing a basis in a vector space. But this is not clear to me. Is there any intuitive idea behind this?

Consider the trivial representation on $V\cong k^n$ with $k$ infinite and $n>1$. Writing $V$ as a sum of irreducibles means finding one-dimensional subspaces $U_1,\cdots,U_n$ of $V$ such that $V$ is the internal direct sum of the $U_i$. Of course there is an obvious way to do this: select the subspaces generated by the coordinate vectors under some assumed basis. However (say $V=k^n$ so the coordinate vectors are our assumed basis) this hardly exhausts all the ways of writing $V$ in this way.

For instance with $n=2$ we could write $V=k(1,0)\oplus k(0,1)$, or we could instead have a different decomposition $V=k(1,1)\oplus k(1,-1)$, which is completely different. In general, each choice of some decomposition $V=U_1\oplus U_2$ corresponds to a choice of two distinct lines through the origin, and there are infinitely many pairs of distinct lines through the origin.

Now consider $U$ an arbitrary irrep of $G$. Since its only subreps are $0$ and $U$, every $u\in U\setminus\{0\}$ spans all of $U$ (as a $k[G]$-module). Let $V=U_1\oplus U_2$ with each $U_1,U_2\cong U$ and $U_i=\langle u_i\rangle$. Just as before, there is a distinct decomposition $V=k[G](u_1+u_2)\oplus k[G](u_1-u_2)$.

In general, if $V=U_1\oplus\cdots\oplus U_n=W_1\oplus\cdots\oplus W_n$ are two distinct decompositions of $V$, with each $U_i,W_i\cong U$ (only one irrep at play again for simplicity), then gluing isomorphisms $U_i\cong W_i$ together we obtain a map $A:V\to V$ such that the image of $V_i$ is $W_i$ as $k[G]$-submodules of $V$: thus we conclude that every decomposition of $V$ can be written as $V=AU_1\oplus\cdots AU_n$ for various automorphisms $A\in{\rm Aut}_G(V)$ and a given decomposition $V=\bigoplus U_i$.

Schur's lemma tells us that $A\in{\rm Aut}_G(V)$, where $V$ has a given decomposition, comes in the form of a block matrix with each block a scalar multiple of an identity matrix. Thus, the ways of rewriting $V$ from one internal decomposition to another are in natural correspondence with the ways of rewriting $k^n$ from its standard decomposition into other decompositions.