Reflection Group of Type $C_n$

Question

In Humphreys' book Reflection Groups and Coexeter groups, he defines a group of type $B_n$ (for $n \ge 2$) in the following way:

Let $V = \Bbb{R}^n$, and define $\Phi$ to be the set of all vectors of squared length $1$ or $2$ in the standard lattice. So $\Phi$ consists of the $2n$ short roots $\pm e_i$ and the $2n(n-1)$ long roots $\pm e_i \pm e_j$ for $i < j$, totalling $2n^2$. For $\Delta$ take $\alpha_1 = e_1 - e_2$, $\alpha_2 = e_2 - e_3$,...., $\alpha_{n-1} = e_{n-1}-e_n$, $\alpha_n = e_n$. Then $\widetilde{\alpha} = e_1 + e_2$ [is the longest root]. $W$ is the semidirect product of $S_n$ (which permutes the $e_i$) and $(\Bbb{Z}/2 \Bbb{Z})^n$ (acting by sign changes on the $e_i$), the latter normal in $W$.

That all good and fine, and the group of type $B_n$ is explicitly defined as a semidirect product. However, the definition for a group of type $C_n$ doesn't appear to be explicitly defined or so it seems:

Starting with $B_n$, one can defined $C_n$ to be its inverse root system (Note that $B_2$ and $C_2$ are isomorphic). It consists of the $2n$ long roots $\pm 2 e_i$ and the $2n(n-1)$ short roots $\pm e_i \pm e_j$ for $i < j$. For $\Delta$, take $\alpha_1 = e_1 - e_2$, $\alpha_2 = e_2 - e_3$,..., $\alpha_{n-1} = e_{n-1} - e_n$, $\alpha_n = 2 e_n$. Then $\widetilde{\alpha} = 2e_1$

So, what exactly does the group of type $C_n$ look like?

Answer 1

A root system $Φ$ in a euclidean vector space $V$ and its dual root system $Φ^∨$ have the same Weyl group:

For any root $α$ of $Φ$, its coroot is given by $$ α^∨ = \frac{2 α}{ (α, α) } \,. $$ This entails that $α$ and $α^∨$ are linearly dependent. The resulting reflections $s_α$ and $s_{α^∨}$ are therefore equal. The Weyl group of Φ is defined as the subgroup of $\operatorname{GL}(V)$ generated by the reflections $s_α$ with $α ∈ Φ$. Similarly for $Φ^∨$. It follows from the equality of sets $$ \{ s_α \mid α ∈ Φ \} = \{ s_{α^∨} \mid α^∨ ∈ Φ^∨ \} $$ that the generated subgroups of $\operatorname{GL}(V)$ are the same, i.e., that the Wey groups of $Φ$ and $Φ^∨$ are the same.

You write that “the group of type $B_n$ is explicitly defined as a semidirect product”. This is wrong. The group of type $B_n$ in not defined as a semidirect product, but instead as the Weyl group of the root system $B_n$. (It then turns out that this group can be described as a semidirect product.)

The group of type $C_n$ is similarly defined as the Weyl group of the root system $C_n$. But the root system $C_n$ is defined as the dual root system to $B_n$, whence two root systems $B_n$ and $C_n$ have the same Weyl group. The groups of type $B_n$ and of type $C_n$ are therefore one and the same.