Why unitary representation?

I am involved in a research project involving representation theory. While we were working through the proofs, one of the project partners asked, why is that we care that the representation of finite groups are unitary?

There are two answers to this question. One is straightforward one. We are interested in characterizing irreducible representations of abstract groups, and their construction involves two theorems. The first theorem says that every representation of a finite group is equivalent to a unitary representation. The second theorem says that any unitary representation of a finite group is either irreducible or decomposable. Using these, we prove the foundational Maschke theorem that says that every representation of a finite group is completely reducible. Hence, the dry answer to our question is that we need the unitary group as a tool in our larger goal of finding decompositions of representations.

But perhaps, we can ask is why is that representation are equivalent to the unitary group and none other? What's special about the unitary group? To understand this, we need to go back to the fundamentals of groups. There is the set-theoretic definition of groups, which is how a group is set and an operation with certain properties. But the other more concrete definition is that a group is the complete set of transformation that preserves some properties of some set of objects (say the shape of a <{http://www.w3.org/1998/Math/MathML}math><{http://www.w3.org/1998/Math/MathML}semantics><{http://www.w3.org/1998/Math/MathML}mrow><{http://www.w3.org/1998/Math/MathML}mi>n<{http://www.w3.org/1998/Math/MathML}annotation encoding="application/x-tex">n-sided regular vertex-labeled polygons). In this latter definition, each group element transforms each of the objects (say the equilateral triangle ABC) to one of other objects (say the equilateral triangle BAC). One essential property here is that we can start with one of the objects and apply any combination of group operations but we never transform to an object outside our original set of objects (otherwise, we didn't construct the group correctly).

Within this picture, when we construct the representation of the group, the group elements get mapped onto linear transformations, and the objects get mapped onto vectors (upon which the linear transformations act), which I will call <{http://www.w3.org/1998/Math/MathML}math><{http://www.w3.org/1998/Math/MathML}semantics><{http://www.w3.org/1998/Math/MathML}mrow><{http://www.w3.org/1998/Math/MathML}mo stretchy="false">{<{http://www.w3.org/1998/Math/MathML}msub><{http://www.w3.org/1998/Math/MathML}mi>v<{http://www.w3.org/1998/Math/MathML}mi>i<{http://www.w3.org/1998/Math/MathML}mo stretchy="false">}<{http://www.w3.org/1998/Math/MathML}annotation encoding="application/x-tex">{v_i}.

Let's think about these vectors in a bit more detail. No element in the object set is special, because the only difference is in the labeling of each object. So, the vectors themselves must reflect this fact. It should be obvious with some thought that we expect the vectors to all have the same length and pairwise angle. That way, relabeling of the objects doesn't discriminate between the various vectors. We can specifically note that for the regular representation the vectors will be orthogonal, but for any smaller representation (dimension smaller than the degree of the group), the vectors corresponding to different objects will not be orthogonal. Moreover, if all the lengths are the same, we can just rescale to set them to one.

I will also note that from the geometric or physical point of view, we are not interested in the vector space spanned by these vectors, because sums of vectors are not well-defined geometrically/physically. If <{http://www.w3.org/1998/Math/MathML}math><{http://www.w3.org/1998/Math/MathML}semantics><{http://www.w3.org/1998/Math/MathML}mrow><{http://www.w3.org/1998/Math/MathML}msub><{http://www.w3.org/1998/Math/MathML}mi>v<{http://www.w3.org/1998/Math/MathML}mn>1<{http://www.w3.org/1998/Math/MathML}annotation encoding="application/x-tex">v_1 is the triangle ABC and <{http://www.w3.org/1998/Math/MathML}math><{http://www.w3.org/1998/Math/MathML}semantics><{http://www.w3.org/1998/Math/MathML}mrow><{http://www.w3.org/1998/Math/MathML}msub><{http://www.w3.org/1998/Math/MathML}mi>v<{http://www.w3.org/1998/Math/MathML}mn>2<{http://www.w3.org/1998/Math/MathML}annotation encoding="application/x-tex">v_2 is the triangle BAC, what does <{http://www.w3.org/1998/Math/MathML}math><{http://www.w3.org/1998/Math/MathML}semantics><{http://www.w3.org/1998/Math/MathML}mrow><{http://www.w3.org/1998/Math/MathML}msub><{http://www.w3.org/1998/Math/MathML}mi>v<{http://www.w3.org/1998/Math/MathML}mn>1<{http://www.w3.org/1998/Math/MathML}mo>+<{http://www.w3.org/1998/Math/MathML}msub><{http://www.w3.org/1998/Math/MathML}mi>v<{http://www.w3.org/1998/Math/MathML}mn>2<{http://www.w3.org/1998/Math/MathML}annotation encoding="application/x-tex">v_1+v_2 mean? Nothing naturally!

Hence, for the construction of the representation, all we need is that each group element/linear transformation be designed in a way so that it maps each vector <{http://www.w3.org/1998/Math/MathML}math><{http://www.w3.org/1998/Math/MathML}semantics><{http://www.w3.org/1998/Math/MathML}mrow><{http://www.w3.org/1998/Math/MathML}msub><{http://www.w3.org/1998/Math/MathML}mi>v<{http://www.w3.org/1998/Math/MathML}mi>j<{http://www.w3.org/1998/Math/MathML}annotation encoding="application/x-tex">v_j onto a unique vector <{http://www.w3.org/1998/Math/MathML}math><{http://www.w3.org/1998/Math/MathML}semantics><{http://www.w3.org/1998/Math/MathML}mrow><{http://www.w3.org/1998/Math/MathML}msub><{http://www.w3.org/1998/Math/MathML}mi>v<{http://www.w3.org/1998/Math/MathML}mi>k<{http://www.w3.org/1998/Math/MathML}mo>∈<{http://www.w3.org/1998/Math/MathML}mo stretchy="false">{<{http://www.w3.org/1998/Math/MathML}msub><{http://www.w3.org/1998/Math/MathML}mi>v<{http://www.w3.org/1998/Math/MathML}mi>i<{http://www.w3.org/1998/Math/MathML}mo stretchy="false">}<{http://www.w3.org/1998/Math/MathML}annotation encoding="application/x-tex">v_k \in {v_i}. Only this correctly constructs the group table (no repetitions in each row). In other words, each linear transformation is just swapping the vectors around.

More specifically, we have some set of vectors that all have the same length and pairwise angle, and we want a transformation that preserves this length and these pairwise angle, while swapping them around. This is the same thing as saying that we want transformations that preserve the inner product between vectors through the transformation: the length is one, so the only degree of freedom in the inner product is the angle, and all pairwise angles are equal. What set of linear transformations preserves the inner product? The unitary group. Hence, every representation of a group must be equivalent to a unitary representation.

Hopefully, this answers the question in an intuitive way that nevertheless can be rigorously formulated.