Why to define representations of unitary groups if they themselves consist of matrices?

Question

I am reading the article called Representations of $U(3)$ in $U(N)$ and there is an example $U(6)$ input irreducible representation $[2,1,1,1,1,0]$, which is then shown to be reduced into $U(3)$ irreps $[6,4,2]$ and $[5,4,3]$. I am new to group theory, but as far as I understand a representation of a group is a homomorphism into a group of matrices with matrix multiplication operation. However, if $U(N)$ itself is a group of matrices, so why to define its representation?

Moreover, what I do not understand is that representation of particular $U(6)$ element should be a matrix, while in the article it is in fact a vector, and the same holds for resulting $U(3)$ irreps.

Rationale: I am a C++ programmer working on the optimization and parallelization of the code that implements the above described reduction. And, I would like to understand the math inside at least a bit, though haven't studied group theory before.

Answer 1

If you have an $n\times n$ matrix, one way to understand it is to decompose $\mathbb{C}^n$ into eigenspaces, if the matrix is diagonalizable. An analogue for a group of invertible matrices is to decompose the vector space into irreducible representations, if the group is semisimple/reductive.

Why define a representation of a matrix group? In applications, representations usually come out of recognizing a symmetry in some pre-existing vector space (for example the solutions to a differential equation), and representation theory aims to be able to give a complete description of any such symmetry that you might come across. Sometimes that existing symmetry is a matrix group, so it is good to be prepared.

A basic example is Fourier analysis of periodic functions. Translation of a periodic function gives $U(1)$ symmetry, and the decomposition of a periodic function along the corresponding irreducible representations is the Fourier transform. (Forgive me for glossing over convergence and such.)

A slightly more involved example is Fourier analysis of functions on the unit sphere in $\mathbb{R}^3$. The sphere has $SO(3)$ symmetry, and the symmetry can be used to decompose such functions into elements of odd-dimensional irreducible representations. Spherical harmonics are a certain basis of these vector spaces and have connections to electron orbitals.

Answer 2

Representations are not there just for the practical purposes that you think of when you hear "matrices", they are above all a way to shed some light onto an algebraic structure by relating it to another one via an action.

As for the "vectors", they only look like vectors but they really are partitions, that is sets of integers that sum up to a particular value. You can find more examples on how this works in the following paper (I couldn't open yours so I don't know how they explain it):

https://arxiv.org/pdf/1405.2169.pdf