There are quite a few basic things I feel I am unfamiliar with here. I want to better understand weights of a representation of a Lie group. The example I'm working through is $SU(n)$.
From what I understand (which may be wrong), we first need to find the fundamental weights from a basis of simple roots. This step is independent of our choice of representation. The next step is where I'm unclear, actually specifying the weights of a representation, not just the fundamental weights.
So far I have that the roots are of the form $\theta_i -\theta_j, 1\leq i < j \leq n$, where $\theta_i$ acts on a diagonal matrix with trace 0, which is an element of $Lie(T)$, $T$ a maximal torus of $SU(n)$, by mapping it to the $i^{th}$ diagonal entry. A natural choice of simple roots are those of the form $\theta_i - \theta_{i+1}$. I then showed, using the Cartan matrix, that the fundamental weights are $w_i = \theta_1 + \ldots + \theta_i$.
Now for the simple case of choosing $\mathbb{C}^n$ with the standard inclusion of $SU(n)$ into $Aut(\mathbb{C}^n)$ to be our representation, I could find online that the weights are simply $\{ \theta_i : 1 \leq i \leq n \}$, with $\theta_n = -\sum_{k=1}^{n-1}\theta_k$. This makes sense, but I can't seem to justify it. What am I missing? Do I even have the right steps so far?