# Reflection Group of Type $C_n$

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?

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.