I have the following set:
$ \text{Pol}(O(n)) := \{ \rho_X | \rho_X(Y) := \text{tr}(X^T Y) \text{ where } X,Y \in O(n)\} $
So, for a function the $X$ is always fixed and we just have the $Y$ as input.
I want to determine the dimension of this polynomial space.
My first question is: do the $X$ play a role in the dimension? Because I have just ignored them and I am not sure whether that is right.
Since I didn't really know how to solve this, I just looked at $O(2)$ at first. We know:
$ O(2) = \Big\{ \begin{pmatrix} \cos a & -\sin a \\ \sin a & \cos a \\ \end{pmatrix}, \begin{pmatrix} \cos a & \sin a \\ \sin a & -\cos a \\ \end{pmatrix} \mid a\in \mathbb{R} \Big\} $
Let $X\in O(2)$ be arbitrary. We write:
$ X = \begin{pmatrix} x_{11} & x_{12} \\ x_{12} & x_{22} \\ \end{pmatrix}$
If we want to look at what $\rho_X(Y)$ is, we can distinguish between two cases for $Y$:
Case 1:$Y = \begin{pmatrix} \cos a & -\sin a \\ \sin a & \cos a \\ \end{pmatrix}$ for some $a\in \mathbb{R}$. Then:
$ \rho_X(Y) = tr \begin{pmatrix} x_{11}\cos a - x_{12} \sin a & \\ & x_{21} \sin a + x_{22} \cos a \\ \end{pmatrix} = \cos a(x_{11} + x_{22}) + \sin a(-x_{12} + x_{21})$
Case 2: $Y = \begin{pmatrix} \cos a & \sin a \\ \sin a & -\cos a \\ \end{pmatrix}$ for some $a\in \mathbb{R}$. Then:
$ \rho_X(Y) = tr \begin{pmatrix} x_{11}\cos a + x_{12} \sin a & \\ & x_{21} \sin a - x_{22} \cos a \\ \end{pmatrix} = \cos a(x_{11} - x_{22}) + \sin a(x_{12} + x_{21})$
Therefore, the only difference is that we have $\pm \sin a x_{12}$ and $\pm \cos a x_{22}$. But if we have been given a $Y$ that is of the form in the first case and we have a $Y'$ that is of the form in the second case, we can also represent $\rho_X(Y')$ with $\rho_{X'}(Y)$ for an $X'$ of the following form:
$ X' = \begin{pmatrix} x_{11} & -x_{12} \\ x_{12} & -x_{22} \\ \end{pmatrix} $
Therefore, the polynomial space has dimension 1.
Would that be correct? If not, can somebody give me a hint?
Also, how would you go on now for $n\geq 2$ and determine the dimension of the polynomial space?
Thanks in advance for any reply!
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Also: If you want to know where I have this exercise from: Godsil's book "Algebraic Combinatorics" (ch. 14, ex. 1)