Mathematics
The Regular Representation of a Finite Group over the Complex Numbers
Quick fact
The regular representation of a finite group over the complex numbers decomposes as a direct sum of all irreducible representations, with each irreducible representation appearing a number of times equal to its dimension. Consequently, the dimension of the regular representation equals the order of the group, and the sum of the squares of the dimensions of all irreducible representations equals the group order.
Why this is interesting
Imagine taking a group of symmetries and turning each symmetry into a matrix that shuffles the elements of the group itself. What does that simple shuffle reveal about the group's hidden structure?