Exercise 4, § 7

Let E be a complex vector space, and let $\mathbf{T}^p_q(E)$ be the space of tensors of type $(p, q)$ on E (Algebra, Chap. III, §5, no. 6). Denote by $H^p_q(E)$ the subspace of $\mathbf{T}^p_q(E)$ consisting of the symmetric tensors (that is, those belonging to the image of $\mathbf{T}\mathbf{S}^p(E)\otimes \mathbf{T}\mathbf{S}^q(E^*)$ in $T^p_q(E)$ and annihilated by the contractions $c^i_j$ for $i\in 1, p$ and $j\in p+ 1, p+q$ (loc. cit.).

a) Show that $H^p_q(\mathbf{C}^n)$ is stable under $\mathbf{G}\mathbf{L}(n,\mathbf{C})$, and consequently under $\mathbf{S}\mathbf{U}(n,\mathbf{C})$; denote this representation of $\mathbf{S}\mathbf{U}(n,\mathbf{C})$ by $\tau_q^p$. b) Show that $\tau_q^p$ is an irreducible representation whose highest weight (with the notations of Exerc. 1 of §4) is $p\varpi_1+q\varpi_{n-1}$.

c) Every irreducible representation of $\mathbf{S}\mathbf{U}(3,\mathbf{C})$ is isomorphic to one of the representations $\tau_q^p$.

d) Let $(x_i)_{1\leq i\leq n}$ be the canonical basis of $\mathbf{C}^n, (y_j)_{1\leq j\leq n}$ the dual basis. Show that $H^p_q(\mathbf{C}^n)$ can be identified with the space of polynomials $P\in \mathbf{C}[x_1, . . . , x_n, y_1, . . . , y_n]$, homogeneous of degree $p$ in the $x_i$ and of degree

$q$ in the $y_i$, and such that $\sum_{i=1}^n\frac{\partial^2P}{\partial x_i\partial y_i}= 0$.