Let $T$ be a tautological representation of Lie algebra $\mathfrak{gl}(N,\mathbb{C})$ in $\mathbb{C}^N$. My goal is to understand why the corresponding representation $T_k$ in symmetric power $\operatorname{Sym}^k\mathbb{C}^N$ is irreducible and find its highest weight.
I found exactly the similar question with the answer, but it details are unclear to me. Can anybody explain the following part?
Now start with a monomial $m_n$ in $W$. If $n$ is not supported by a single element, applying a suitable unipotent element, we can increase the value of some element. Iterating, we see that $W$ contains a monomial of the form $x^k_i$, and using permutations matrices, it contains all such monomials.
Again using unipotent elements, we see it contains all $x^ℓ_ix^{k−ℓ}_j$, and so on, we eventually get all monomials. (Argue by induction on $n$: then we have all monomials with $n_n=0$; then we get $m=∏_{i<n}x^{n_i}_ix^s_n$ by induction on $s$: apply a unipotent matrix to the monomial $mx_1/x_n$).
I would be very glad to see it on clear examples if it possible (for example, maybe it is possible to show this in case of $\mathfrak{gl}(2,\mathbb{C})$ or $\mathfrak{gl}(3,\mathbb{C})$ and symmetric power $2$ or $3$).
As for the highest weight it should be the maximal eigenvalue of diagonal matrices from $\mathfrak{gl}(N, \mathbb{C})$, am I right? But how can I find it? It even seems to me that it is infinite (because I always can choose matrix with greater elements on diagonale).
Thanks in advance.