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?
Read the full explanation
Understanding The Regular Representation of a Finite Group over the Complex Numbers
Think of a finite group, like the symmetries of a square, as a list of its elements. Now, for each element g, we can define a linear operation that takes the entire list and permutes it: the operation sends each element h to gh. This is the left regular representation. At first, it just looks like a fancy way to reorder a list, but because we are working over the complex numbers, we can form linear combinations of the list entries. That turns the list into a complex vector space, and each group element becomes a matrix that tells how the vector space is shuffled. The key idea is that this representation, built from nothing more than the group's multiplication table, actually contains every possible way the group can act linearly on a complex vector space. In other words, it is a 'universal' representation. For example, if you take the symmetric group S3 (the symmetries of a triangle), the regular representation is 6-dimensional, and it contains the trivial representation, the sign representation (which sends even permutations to 1 and odd to -1), and the standard 2-dimensional representation that rotates and reflects the triangle. Each of those appears as a 'building block' inside the regular representation.
A deeper explanation
The mechanism behind the regular representation is the action of the group on itself by left multiplication. Formally, if G is a finite group of order n, we consider the vector space V with basis {eg : g in G}. Then the left regular representation is defined by ρ(g)(eh) = e{gh}. Because the group operation is associative, this is a genuine representation. Over the complex numbers, the group algebra C[G] is semisimple by Maschke's theorem, so V decomposes as a direct sum of irreducible subrepresentations. The crucial fact is that every irreducible representation of G appears in this decomposition, and its multiplicity equals its dimension. This is seen by computing the character of the regular representation: χreg(g) = n if g = e (the identity) and 0 otherwise. Using the orthogonality of characters, the multiplicity of an irreducible character χ in the regular representation is ⟨χreg, χ⟩ = χ(e) = dim(χ). This result implies the fundamental identity |G| = Σi (dim Vi)^2, where the sum runs over all irreducible representations. This not only gives a way to list irreducible representations but also underpins the character table and the orthogonality relations. The regular representation thus serves as a bridge: it is a concrete, always available representation that encodes the full representation theory of the group.