representation $\pi_{m,\,n}: \text{SU}(n) \to \text{GL}(V_m)$

Question

Let $V_{m,\,n}$ denote the vector space of the homogeneous complex polynomials of degree $m$ in $n$ variables (under addition).

1. Define a representation $\pi_{m,\,n}: \text{SU}(n) \to \text{GL}(V_m)$. Prove that it is continuous.
2. What is the degree of $\pi_{m,\,n}$, i.e. what is the dimension of $V_{m,\,n}$?
3. For which $\pi_{m,\,n}$ does there exist an $\text{SU}(n)$-invariant vector?

Progress so far: I understand for the first part how to do the case where $m = 3$. But I am not sure how to extend it to general $m$. Any help would be appreciated.

Answer 1

1.

It suffices to show that the action of ${SU}(n)$ on a single basis polynomial ${x_1}^{a_1} {x_2}^{a_2} \dots {x_n}^{a_n}$ $($where $a_1 + \dots + a_n = m$$)$ is continuous. That is, let $A = (a_{ij}) \in {SU}(n)$ be a matrix, then we need to show that the change of variables $$(x_1, x_2, \dots, x_n) \mapsto \left(\sum a_{1i} x_i, \sum a_{2i} x_i, \dots, \sum a_{ni} x_i \right)$$ produces small changes in its effect on ${x_1}^{a_1} {x_2}^{a_2} \dots {x_n}^{a_n}$ if the entries $(a_{ij})$ are perturbed slightly. But when the polynomial $A({x_1}^{a_1} {x_2}^{a_2} \dots {x_n}^{a_n})$ is expanded out, every coefficient is a polynomial in the $a_{ij}$, i.e. a continuous function of the $a_{ij}$. Hence small perturbations in the $a_{ij}$ lead to small perturbations in $A({x_1}^{a_1} {x_2}^{a_2} \dots {x_n}^{a_n})$. Thus, the representation is continuous.

2.

We need to count the number of basis polynomials ${x_1}^{a_1} {x_2}^{a_2} \dots {x_n}^{a_n}$ where $a_1 + \dots + a_n = m$. This amounts to counting the number of nonnegative integer solutions to $a_1 + \dots + a_n = m$. Suppose we have $n-1$ black balls and $m$ white balls in a row. Every manner in which we can arrange this row corresponds bijectively to a solution of $a_1 + \dots + a_n = m$, where the black balls are dividers between portions of the partition. The number of ways to make this arrangement is $$\binom{m+n-1}{m},$$ so this is the dimension of $V_m$.

3.

Let $$P = \sum_{a_1 + \dots + a_n = m, \, s = (a_1, \dots, a_n)} c_s {x_1}^{a_1} \dots {x_n}^{a_n}.$$ If this is invariant under ${SU}(n)$, then it is invariant under diagonal elements, so for any $z_1, \dots, z_n \in S^1$ with $z_1 \dots z_n = 1$, we have ${z_1}^{a_1} \dots {z_n}^{a_n} = 1$ $($where $a_1, \dots, a_n$ is an arbitrary solution to $a_1 + \dots + a_n$$)$. Taking the solution where exactly one $a_i$ is nonzero, this implies every $z_i$ is an $m$th root of unity. But there exist $n$-tuples $(z_1, \dots, z_n)$ with $z_1 \dots z_n =1$ where not all of the $z_i$ are roots of unity $($exercise left to the reader to come up with an example$)$. Hence, there are no invariant polynomials.