Irreducible representation

Okay, let's imagine we are playing with a bunch of toys, and we want to find out how they can move around in different ways. Imagine we have a big toy car that can move forward, sideways, and backward. And we also have a little toy car that can only move forward and backward.

Now, we want to understand how these toys move based on some rules. And to do that, we need to group these toys based on how they are similar. For instance, we can group the big toy car and the little toy car together since they can both move forward and backward.

Such groups are called "representations." And when we talk about "irreducible representation," we mean that we cannot split a group of toys into smaller subgroups that would still obey the same rules. It is like trying to break a big toy into smaller pieces, but each piece still has to move the same way as the original big toy.

In other words, an irreducible representation is like a building block that cannot be broken down into smaller pieces that obey the same rules. It is a fundamental way of understanding how a bunch of similar toys move based on some rules, and we cannot make it any simpler than it already is.