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.
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Sign up to join this communityAre 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.
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.