I am reading a paper to obtain a joint probability density function of a particular random variables. In the paper, the authors employ representation theory of unitary groups, which I have no background at all. Thus, I would like to ask for some enlightenments on how to explicitly obtain the values of certain variables, if possible.
According to the authors, a $d$-dimensional representation of a group $G$ is a homomorphism from $G$ into a group of $d$-dimensional invertible matrices. Such group include $Gl(M)$, the group of complex invertible matrices of dimension $M$, and $U(M)$ is its subgroup of unitary matrices.
A $d$-dimensional representation is irreducible if it has no non-trivial invariant subspaces. In other words, a representation is irreducible if there exists no $d$-dimensional invertible matrix $A$ such that $AHA^{-1}$ becomes a block diagonal for all group elements $H$.
The irreducible representations of the unitary group $U(M)$ (and $Gl(M)$) can be labeled by an $M$-dimensional vector $m=[m_1,m_2,\cdots,m_M]$ with integers $m_1\geq m_2 \geq \cdots \geq m_M \geq 0$.
My question is can I actually compute the vector $m$ explicitly if I am given an (or a set of) arbitrary unitary matrix(matrices)? For example, $m_1=3,m_2=2,\cdots$ etc. Or should I abstractly assume that such vector $m$ exists?
Through some videos and books, I assume is the latter. Because what I roughly understand from this theory is that we can ignore the context of the matrix as long as it falls into the group. Since it falls into the group, any relevant formula (e.g., Orthogonality Relation Between Unitary Group Matrix) follows through. But this seems impossible because the probability density function is also given as a function of $m$. So, one should be able to compute $m$ explicitly. Kindly advice.
Besides, I think I don't really understand the keywords (e.g., homomorphism, representation/irreducible). Is there any easy-to-understand chapter/notes/videos/examples that I can refer to?
You can’t deduce the label $m$ from just one matrix. As an extreme example, all representations map the identity element of the group to the identity matrix, so you can’t tell which representation you have by looking at just one of its matrices for one group element. If you have the entire representation (which is neither a matrix, nor a set of matrices, but a function that assigns a matrix to every group element), you can determine its irreducible components using something called character theory.
The character of a representation $\rho$ is the group function that maps each group element $g$ to $\operatorname{Tr}\rho(g)$, the trace of its matrix under the representation. These group functions are class functions, meaning that they’re constant on conjugacy classes because the trace is cyclically invariant:
$$\operatorname{Tr}\rho(a^{-1}ga)=\operatorname{Tr}\rho(a)^{-1}\rho(g)\rho(a)=\operatorname{Tr}\rho(g)\rho(a)\rho(a)^{-1}=\operatorname{Tr}\rho(g)\;.\;$$
(By the way, to clear up one of those mysterious “keywords”: $\rho$ being a homomorphism basically means that we can do what I just did: move all the operations (multiplication and inversion) from the group elements to the matrices – this is often expressed as “a homomorphism is a function that respects the group structure”.)
It turns out that the characters of the irreducible representations span the vector space of class functions and are orthonormal with respect to a suitable inner product, so they form an orthonormal basis. The character of a direct sum of representations is just the sum of their characters, so you can figure out which irreducible components a given representation decomposes into by projecting its character onto the characters of the irreducible representations.
The representations of the unitary groups are a relatively complicated example for your first encounter with representations and characters. If you really need this for what you’re doing, I’d suggest to start with some examples of small finite groups. For instance, the symmetric group $S_3$ has three conjugacy classes and thus three irreducible representations, two one-dimensional and one two-dimensional, that you can more easily get a handle on. When you feel you’ve understood representations and characters of finite groups, you can move on to $U(1)$, whose representation theory is quite a bit easier than that of the higher unitary groups. Once you’ve mastered that, you’ll be prepared to take on the $m$ vectors :-)
Regarding easy-to-understand materials: I studied physics, and I found Group theory and physics by Sternberg very useful – but I can’t tell how useful it will be to you with a background in computer science.
If you link to the paper, I might be able to say more about what might help you understand it.