# Let $T$ be a maximal torus of $SU(k+1)$. Who is the normalizer $N(T)$ of $T$ in $O(2k+2)$?

I'm reading an article which I cannot understand a paragraph very well.

$$T$$ is a maximal torus of $$SU(k+1)$$ acting linearly on $$\mathbb{C}^{k+1}$$. And here is what is written that I cannot fully understand:

Note that the normalizer $$N=N(T)$$ of $$T$$ in $$O(2k+2)$$ normalizes the centralizer $$C(T)$$ of $$T$$ in $$O(2k+2)$$. Assume $$k\geq 2$$. Since $$C(T)$$ is a maximal torus of $$U(k+1)$$ which is also a maximal torus of $$O(2k+2)$$, we see that $$C(T)$$ is the identity component of the normalizer $$N$$. Moreover, $$N$$ is generated by $$C(T)$$ , the complex conjugation $$c$$ and the symmetric group $$S_{k+1}$$ of permutations of complex coordinates.

First of all, I cannot see why $$C(T)$$ should be a maximal torus of $$U(k+1)$$. I can see why it should be connect, since it is the union of maximal tori in $$U(k+1)$$ containing $$T$$. My supervisor also said that it is because a element in $$C(T)$$ lies on $$N(T)$$, and we can assume, using complex conjugation $$c$$, that every element $$n\in N(T)$$ is complex-linear on $$\mathbb{C}^{k+1}$$ (and I cannot see why). And that, up to permutation, $$n$$ preserves the decomposition $$\mathbb{C}^{k+1}=\mathbb{C}\oplus\cdots\oplus \mathbb{C}$$ (which I also cannot see why). And furhter he said that such an element must Lie on the maximal torus $$S^1\times\cdots\times S^1\subset U(K+1)$$; factorization compatible with the decomposition $$\mathbb{C}^{k+1}=\mathbb{C}\oplus\cdots\oplus \mathbb{C}$$ (and, again, I cannot see why).

Given this, why should $$C(T)$$ be the indentity component of $$N(T)$$?

• Why don't you calculate? Take $k=2$. Take the diagonal maximal torus in ${\rm SU}(3)$; it is 2-dimensional. Compute its centralizer in $U(3)$; it is the 3-dimensional diagonal torus. Clearly this 3-dimensional torus is a maximal torus in $U(3)$ and a maximal torus in ${\rm SO}(6)$. You continue calculations.... Commented Jul 5, 2020 at 10:36
• I have calculated it to the usual torus in SU(n), but I would like to understant it more geometrically/generally, for an arbitrary maximal torus of $SU(n)$. I know that all need can be stracted from those calculations, but I would like to understand the techinics used by the author, which are much more general. Commented Jul 5, 2020 at 19:44
• It is the same! You take an arbitrary maximal torus $T\subset {\rm SU}(n)$, you consider its natural complex representation in $V:={\Bbb C}^n$, this representation uniquely decomposes in a direct sum of one-dimensional complex representations, you take eigenvectors as basis vectors, and you get the "usual" diagonal torus. Then you should calculate! Commented Jul 6, 2020 at 8:32