# Type $C_n$ Weyl group contains in the centralizer of the longest word $w_0$ in $S_{2n}$

Are there some references about the proof of the following fact?

Type $$C_n$$ Weyl group lies in the centralizer of the longest word $$w_0$$ in $$S_{2n}$$.

Thank you very much.

• So you take the longest element $w_0$ in $S_{2n}$ and want to show that there is a subgroup of the centralizer $C_{S_{2n}}(w_0)$ that is isomorphic to $C_n$? Why do you call it a Weyl group? Is it important or are you ok with any $C_n$ that could be found in the centralizer?
– Dirk
Oct 5, 2017 at 8:53
• The longest word is $(1,2n)(2,2n-1) \ldots (n,n+1)$ of cycle type $(2^n)$. So its centralizer is $C_2 \wr S_n$, which is the Weyl group of type $B$ or $C$. Reference: any textbook on Lie theory / Coxeter groups, plus e.g. James and Kerber, The representation theory of the symmetric groups, Chapter 4. Oct 5, 2017 at 10:22
• Small linguistic question: does your expression "contains in" (in the header and question) actually mean "lies in"? (Also, a tag 'reference-request' is useful here.) Oct 5, 2017 at 14:02
• @JimHumphreys, thank you very much. Yes, I need to use "lies in". I will add a tag "reference-request". Oct 5, 2017 at 15:18

Although the comment of Mark Wildon certainly addresses the question, here is another (maybe more root system-y) perspective. Conjugation by $$w_0$$ in $$S_{2n}$$ corresponds to the non-identity involutive automorphism of the Dynkin diagram of Type $$A_{2n-1}$$ (i.e., reflect the diagram across its vertical axis of symmetry). The centralizer of $$w_0$$ is exactly the fixed point subgroup of this automorphism. And in general, for any diagram automorphism, the fixed point subgroup is naturally isomorphic to the Weyl group of the "folded" diagram: in this case the folded diagram is the Type $$B_{n}$$ diagram. See Stembridge's write-up on folding (http://www.math.lsa.umich.edu/~jrs/papers/folding.pdf), in particular, Claim 3 there.