Calculating a Representation of Polynomials

Question

I'm just starting to learn representation theory and am having trouble with a computation. This shouldn't be tricky at all, I'm just having trouble interpreting one thing:

Let $\mathcal{P}^n_d$ be the vector space of real valued polynomials of $n$ variables and $deg(p) \leq d$. For any $A\in GL_n(\mathbb{R})$ we can define a map $\tau_d^n(A):\mathcal{P}_d^n\rightarrow\mathcal{P}_d^n$ by $\tau_d^n(A)p=p\circ A^{-1}$, thus we have $\tau_d^n:GL_n(\mathbb{R})\rightarrow GL(\mathcal{P}^n_d)$. The goal is to prove that this is a faithful representation, which I am was able to do. However, I was curious what $\tau_d^n(A)p$ actually looks like.

I took the example $A=\begin{bmatrix}1 & 1 &2\\1 & 2 & 1\\ 2 & 1 & 1 \end{bmatrix}$, whose inverse is $A^{-1}=\frac{1}{4}\begin{bmatrix}-1 & -1 & 3\\ -1 & 3 & -1\\ 3 & -1 & -1 \end{bmatrix}$. I also took $p\in\mathcal{P}_d^n$ to be $p=x_1+x_2^2+x_2x_3$.

I'm really not sure how to interpret:

$(x_1+x_2^2+x_2x_3)\circ\frac{1}{4}\begin{bmatrix}-1 & -1 & 3\\ -1 & 3 & -1\\ 3 & -1 & -1 \end{bmatrix}$

Any insight would be appreciated, thanks!

Answer 1

In this setting, it is harmless to identify polynomials in $n$-variables with functions on $\mathbb{R}^n$; $\circ$ simply means composition of functions.

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More generally, given any commutative algebra $S$ over the reals, we can:

• View a real polynomial in $n$ variables as a function $S^n \to S$
• View a real $n \times n$ matrix $A$ as a linear function $S^n \to S^n$

And the polynomial $p \circ A$, when viewed as a function, should be the composites of $p$ and $A$ viewed as functions.

To work out what $p \circ A$ must be, it is convenient to take $S$ to be the algebra of all real polynomials in $n$ variables, and exploit the fact that $p = p(x_1, x_2, \ldots, x_n)$. (on the left, $p$ is a polynomial, on the right $p$ is a function)