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In the following, I denote the space of polynomials of degree $N$ in $\mathbb{R}^3$ as $P(\mathbb{R}^3)$. The subspace of homogeneous polynomials will be called $P_N (\mathbb{R}^3)$. I denote the variables in $\mathbb{R}^3$ as $x_1, x_2, x_3$. Now, consider the usual action of $SO(3)$ in $\mathbb{R}^3$: $$\star : \begin{pmatrix} x_1 \\ x_2 \\ x3 \end{pmatrix} \mapsto O \star \begin{pmatrix} x_1 \\ x_2 \\ x_3 \end{pmatrix},$$

$O \in SO(3)$, i.e. where $\star$ is simply matrix multiplication. This gives me a representation $\pi$ of $SO(3)$ in $P (\mathbb{R}^3)$: $$(\pi (O) p) (x) = p(O^{-1} \star x) = p(O^{-1} x),$$ with $O \in SO(3), p \in P$ and $x \in \mathbb{R}^3$.

The restriction of $\pi$ to $P_N (\mathbb{R}^3)$ is denoted $\pi_N$.

I am asked to compute the character $\chi_N$ of $\pi_N$ for a certain matrix. Specifically, the matrix

$$g = \begin{pmatrix} \cos (\theta) & - \sin (\theta) & 0 \\ \sin (\theta) & \cos (\theta) & 0 \\ 0 & 0 & 1 \end{pmatrix} \in SO(3).$$

I know that if I have a fininite-dimensional representation $(\pi, V)$ of some group $G$, then the character $\chi_{\pi}$ of a function $G \rightarrow \mathbb{C}$ defined by $\chi_{\pi} (x) = \text{tr} \pi (x)$, where $\text{tr}$ is just the trace.

But I don't see how I can use this. How do I compute $\chi_N (g) = \chi_{\pi_N}(g) = \text{tr} \pi_N (g)$?

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  • $\begingroup$ Pick a basis for your space; that will let you write a matrix for your group action (the first row is the image of the first basis element, written in the same basis, etc.). Now take the trace. $\endgroup$ – Steve D Jun 16 '18 at 2:13
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Hint (expanding the comment of @SteveD): The Matrix you give has the property that its inverse sends $x_1$ to $\cos(\theta)x_1-\sin(\theta)x_2$, it sends $x_2$ to $\sin(\theta)x_1+\cos(\theta)x_2$ and it leaves $x_3$ fixed. A Basis for $P_N(\mathbb R^3)$ is given by the monomials $x_1^ix_2^jx_3^{N-i-j}$ for $i+j\leq N$. Now compute what $g^{-1}$ does to such a monomial, use this to determine the matrix of your transformation and compute its trace.

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