$V^{\oplus3}$, linear constraints.

Question

Let $V$ be an irreducible $G$-representation over $\mathbb{C}$, and let $W = V \oplus V \oplus V$. Prove that all submodules of $W$ are given by "imposing linear constraints," e.g.$$\{(x, y, z) \in V \oplus V \oplus V: 2x + 3y + z = 0,\, x - y - z = 0\}$$is an example of a submodule obtained thus.

Answer 1

Consider first of all a nonzero irreducible subrepresentation $W \subset V^{\oplus n}$. Each projection map $\pi_j: W \to V$ is either zero or an isomorphism, by Schur's Lemma. Choose $j_0$ so that $\pi_{j_0}$ is an isomorphism. For every $1 \le k \le n$, the map$$\pi_k \circ \pi_{j_0}^{-1}: V \to V$$is, by Schur's Lemma, a scalar; call this scalar $\lambda_k \in \mathbb{C}$. Then, for every $w \in W$,$$w = (\pi_1(w), \pi_2(w), \dots, \pi_n(w)) = (\lambda_1\pi_{j_0}(w), \lambda_2\pi_{j_0}(w), \dots, \lambda_n\pi_{j_0}(w)).$$Since $\pi_{j_0}$ is surjective, we conclude that$$W = \{(\lambda_1v, \lambda_2v, \dots, \lambda_nv)\text{ }|\text{ }v \in V\}.\tag*{(1)}$$This classifies the irreducible subrepresentations of $V^{\oplus n}$. We now consider the general case; for this, we need some notation to handle "systems of equations."

For any subspace $Y \subset \mathbb{C}^n$, let$$W(Y) = \{(v_1, \dots, v_n) \in V^{\oplus n}\text{ }|\text{ }\sum y_iv_i = 0\text{ for all }(y_1, \dots, y_n) \in Y\}.$$Then $W(Y) \subset V^{\oplus n}$ is a subrepresentation; we shall show all subrepresentations arise thus. One has the following dimension formula, the proof of which is left as an exercise to the reader,$$\dim W(S) = (n-\dim S)\dim V.\tag*{(2)}$$Let $W \subset V^{\oplus n}$ be an arbitrary subrepresentation. We may write $W$ as an direct sum $W_1 \oplus \dots \oplus W_k$ where each $W_i$ is irreducible. Each $W_i$ is of the form $(1)$ for some vector $\underline{\lambda}_i = (\lambda_{i1}, \dots, \lambda_{in})$. Let $S = \{{\bf y} \in \mathbb{C}^n\text{ }|\text{ }\underline{\lambda}_i \cdot {\bf y} = 0\text{ for all }1 \le i \le k\}$ $($here we write ${\bf x} \cdot {\bf y} = \sum_j x_jy_j$$)$.

Then $S$ is a subspace of dimension $\ge n - k$, because it is defined by $k$ linear equations. On the other hand, $W_i \subset W(S)$ for each $i$, so also $W \subset W(S)$. But $\dim W = k\dim V $ whereas $(2)$ shows $\dim W(S) \le k\dim V$. So $W = W(S)$.

We have shown that every subrepresentation is of the form $W(S)$ for some $S \subset \mathbb{C}^n$, as required.