Derivatives in a Hilbert space.

Question

We need help with the proof of Lemma IX.11.4 on page 249-250 of the book "Representations of Finite and Compact Groups" by Barry Simon.

The problem has mostly to do with the notation used. We do not understand what he meant with the derivative that is being taken there.

In particular, the theorem says the following:

Let $X$ be a Hilbert space. Let $S_n$ (the symmetric group) act on $X^{\otimes n}$ in the natural way. Let $S^n(X)$ be the set of vectors invariant under all $V_\pi$. Then $S^n(X)$ is the smallest space containing $\{ x \otimes \dots \otimes x \mid x \in X \}$.

Here, the 'natural way' means, for $\pi \in S_n$ and $x_1 \otimes \dots \otimes x_n \in X^{\otimes n}$ that $V_\pi x_1 \otimes \dots \otimes x_n = x_{\pi^{-1}(1)} \otimes \dots \otimes x_{\pi^{-1}(n)}.$

The proof starts with defining $P(x) = x \otimes \dots \otimes x$ and then taking the derivative $$ \left. \frac{\partial}{\partial \lambda_2 \dots \partial \lambda_n} P(e_1 + \lambda_2 e_2 + \dots + \lambda_n e_n) \right|_{\lambda_2 = \dots = \lambda_n = 0} = \sum_{\pi \in S_m} V_\pi(e_1 \otimes \dots \otimes e_n).$$

This derivative is what we don't understand about the proof. We don't know how to actually compute it.

Answer 1

You can expand \begin{align}(e_1 + \lambda_2 e_2 + \dots + \lambda_n e_n )^{\otimes n} &= e_1^{\otimes n}\\ &+e_1^{\otimes(n-1)}\otimes(\lambda_2 e_2 + ... + \lambda_n e_n) \\&+ e_1^{\otimes (n-2)} \otimes\sum_{i=2}^n \sum_{j=2}^n\lambda_i\lambda_j e_i\otimes e_j \\ &+ \dots\\ &+ e_1 \otimes \sum_{i[2]=2}^n\dots\sum_{i[n]=2}^n \lambda_{i[2]}\dots\lambda_{i[n]} e_{i[2]}\otimes\dots\otimes e_{i[n]} \\ &+ (\text{terms without $e_1$})\end{align} The first line vanishes on any derivative in any $\lambda_i$, The second line vanishes on any second derivative, and so on until before the second last line.

The last line consists of terms that have a factor of $\lambda_i^2$ for some $i$. Therefore, their derivative has $\lambda_i$ as a factor, which then vanishes as $\lambda_i\to 0$.

For the second last line, the only terms that do not vanish are the ones for which we have exactly one of each of $\lambda_2,\dots,\lambda_n$. These terms clearly make up a sum over the permutations of $\{2,...,n\}$.

Answer 2

For $n=2$, we have only one variable, $\lambda_2=:t$, and a function $p:\Bbb R\to X, \ t\mapsto P(e_1+te_2)$.

Since the norm on $X$ induces a metric (and topology), we can easily transform the definition of differential for $\Bbb R\to X$ functions: $$f'(t_0):=\lim_{t\to t_0}\frac{f(t)-f(t_0)}{t-t_0}$$ Now we have $p(t)=P(e_1+te_2)=(e_1+te_2)\otimes(e_1+te_2)=(e_1\otimes e_1)+t(e_1\otimes e_2+e_2\otimes e_1)+t^2(e_2\otimes e_2)$
and when differentiating it at $t=0$, the first term vanishes because it's constant, and so does the last term because we evaluate $2t(e_2\otimes e_2)$ at $t=0$.

The multivariate case is analogous: exactly the terms of the form $V_\pi(e_1\otimes\dots\otimes e_n)$ will not vanish.
I suggest to work out the case $n=3$ in detail.