I'm interested in the following statement:
an irreducible representation of dimension $d$ contains $d^2$ linearly independent matrices
that can be found in H. C. Lee (1948), On Clifford Algebras and Their Representations:
Clifford algebras are defined in the following way: given $n$ objects $(u_\lambda)_{\lambda=1}^n$ verifying
$$u_\lambda u_\mu + u_\mu u_\lambda=2\delta_{\lambda\mu}\,,\quad 1\le\lambda,\mu\le n\,,$$
their Clifford algebra is the linear combinations of all their (finite) products (this should be standard, as far as I know). The author is using a generic algebraically closed field with characteristic $\ne 2$, but sticking to complex matrices would be more than fine [edit: initially I forgot the closure request, so I have to apologize].
It seems to me that what, in today's literature, is known as Burnside's Theorem is a different statement (maybe implying the one I'm interested in, but not in a way that can be obvious to me).
I also took a look at the author's reference, van der Waerden, and found this Theorem of Burnside:
In an absolutely irreducible semigroup of matrices of $n$-th degree there are exactly $n^2$ linearly independent matrices
but then the text is invoking several notions, such as of "hypercomplex systems", which I'm not familiar with.
So my question is: is there some simple set of hypotheses under which the statement is true?, if yes, is there a clean and elementary proof of it?
My level of understanding is quite basic, I'm afraid; I know just a bit of representation theory for finite groups, no more than what is covered in the first three chapters of Simon's book, [*] so when we're dealing with a finite-dimensional algebra representation, I just think of it as a unital algebra homomorphism to a complex matrix algebra, and of its reducibility/irreducibility just by borrowing the corresponding notion for finite groups.
[*] indeed, I've been struggling with Simon's fourth chapter, the one on Clifford groups and algebras, precisely because I'm not sure I correctly got his definitions. He seems to be working directly with matrices, in which case imho there is an ambiguity with the order of the group for at least some odd $n$.
This seems like a very old-fashioned book; you'd likely be off reading something more modern. Here is a precise statement: if $A$ is an associative algebra over an algebraically closed field $k$ (this hypothesis cannot be dropped) and $M$ is an irreducible representation / simple module over $A$ of dimension $n$, with action map
$$\rho : A \to \text{End}_k(M) \cong M_n(k)$$
then $\rho$ is surjective, so its image is all of $M_n(k)$ and hence has dimension $n^2$. This follows from Schur's lemma together with the Jacobson density theorem. I am not aware of Burnside's name being attached to this but I suppose it wouldn't be surprising. It can also be deduced from the Artin-Wedderburn theorem (applied to the image of $A$, but here one has to know that two different definitions of "semisimple" for a ring coincide).
If $k$ isn't algebraically closed then the statement is instead that the image of $\rho$ is $\text{End}_D(M)$ where $D = \text{End}_A(M)$ is the division ring of $A$-endomorphisms of $M$, and this can have dimension not a square over $k$; for example if $k = \mathbb{R}$ we could have $D = \mathbb{C}$ which has dimension $2$, and matrix rings over $\mathbb{C}$ would have dimension twice a square over $\mathbb{R}$.