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In particle physics, I was taught that a representation of a group is a function $r: group \rightarrow matrices\,(n\times n)$ such that $r(g_1)r(g_2)=r(g_1g_2)$ and $r(e)=I_{n\times n}$. Then, that a representation is reducible when you can find a matrix $A$ such that $Ar(g)A^{-1}$ is in diagonal-block form for every element of the group.

Then the professor tried to find in complicated ways reducible representations of $SO(N)$, $SU(N)$ and so on. But the trivial function that assigns $I_{n\times n}$ to every value of $g$ is not already a reducible transformation? I know it must be somehow useless, but what did I lose?

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    $\begingroup$ If $n>1$ then yes, the identity representation is trivially reducible and irreducibly trivial. :) You are saying that if you have a bunch of objects which don't transform at all under the group then they don't transform into each other. This is a true statement but we tend to leave such trivial cases out of the discussion. $\endgroup$
    – Michael
    Commented Feb 13, 2013 at 1:31
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    $\begingroup$ Yes that would be a reducible representation decomposed into direct sums of the trivial (singlet) representation $1\oplus\dots\oplus 1$. What do you mean by what you lost? You can construct reducible representations by taking direct sums of irreducible ones in all sort of ways, so this is just the most trivial example of them all (using only the singlet representation). We are typically much more interested in irreducible representations. $\endgroup$
    – Heidar
    Commented Feb 13, 2013 at 1:31

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What you have constructed is a representation, but not a faithful one. Since your homomorphism $r$ is not injective, you lose some of the structure of the group. In fact, since $r$ is trivial, you lose all the structure of the group. While most useful statements about $G$ apply to $r(G)$ equally well, you cannot pull back anything useful from $r(G)$ to $G$, so your representation doesn't tell you anything about $G$, defeating the whole purpose of using representations in the first place.

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Well if you're just looking for one example of a reducible representation of a group, then that's a fine one. However, representation theory of groups has a bunch of wonderful complexity that is illuminated by studying other examples of reducible representations with a lot more structure.

For example, addition of angular momentum in quantum mechanics involves writing reducible representations of $\mathrm{SU}(2)$ as directs sums of irreducible ones.

Cheers!

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