Exercise 5, § 13

The notations are those of no. 1.IV, type $A_l$. Show that, if $n$ is $\geq 0,\mathbf{S}^n\sigma$ is an irreducible representation of $\mathfrak{s}\mathfrak{l}(l+ 1, k)$ of highest weight $n\varpi_1$. (Realise $\mathbf{S}^n\sigma$ on the space of homogeneous polynomials of degree $n$ in $X_0, . . . ,X_l$, and observe that, up to homothety, the only polynomial $f$ such that $\partial f /\partial X_i= 0$ for $i\geq 1$ is $X^n_0.$) Show that all the weights of this representation are of multiplicity 1.