Cn root system ($C_n$)

In an orthonormal basis $({\epsilon }_i)$, the $C_n$ roots are $\pm {\epsilon }_i\pm {\epsilon }_j$ for $i<j$ and $\pm 2{\epsilon }_i$.

For $n\ge 2$, the finite Dynkin diagram has a chain of short roots ${\alpha }_1,\dots ,{\alpha }_{n-1}$ with single bonds, followed by a double bond to the long root ${\alpha }_n$, pointing toward ${\alpha }_{n-1}$. The Extended Dynkin diagram adds the long root ${\alpha }_0=-2{\epsilon }_1$, doubly bonded to ${\alpha }_1$ with arrow toward ${\alpha }_1$. All other bonds remain. For $n=1$, the finite diagram is a single vertex; the affine diagram has two equal-length vertices joined by a double bond without an arrow, with Cartan matrix $\left(\begin{array}{cc}2& -2\\ -2& 2\end{array}\right)$.

Choose positive roots of a root system ${\epsilon }_i-{\epsilon }_j$, ${\epsilon }_i+{\epsilon }_j$ for $i<j$, and $2{\epsilon }_i$. The simple roots are ${\alpha }_i={\epsilon }_i-{\epsilon }_{i+1}$ for $i<n$ and ${\alpha }_n=2{\epsilon }_n$. Their coroots are ${\alpha }_i^{\vee }={\epsilon }_i-{\epsilon }_{i+1}$ and ${\alpha }_n^{\vee }={\epsilon }_n$. Therefore the fundamental weights and Weyl vector areThe highest root is $2{\epsilon }_1=2{\alpha }_1+\cdots +2{\alpha }_{n-1}+{\alpha }_n$. These formulas include rank one.

The Weyl group of the Cn root system is the signed symmetric group. A Weyl reflection in $2{\epsilon }_i$ changes the sign of coordinate $i$; one in ${\epsilon }_i-{\epsilon }_j$ interchanges coordinates $i,j$; one in ${\epsilon }_i+{\epsilon }_j$ interchanges and negates both coordinates. These generate all independent sign changes and all permutations, giving order $2^{n}n!$.

In an orthonormal basis $({\epsilon }_1,{\epsilon }_2)$, the roots are $\pm ({\epsilon }_1-{\epsilon }_2)$, $\pm ({\epsilon }_1+{\epsilon }_2)$, $\pm 2{\epsilon }_1$, and $\pm 2{\epsilon }_2$. Choose simple roots ${\alpha }_1={\epsilon }_1-{\epsilon }_2$ and ${\alpha }_2=2{\epsilon }_2$, with ${\alpha }_1$ a short root. The fundamental weights are ${\omega }_1={\epsilon }_1$ and ${\omega }_2={\epsilon }_1+{\epsilon }_2$.

The $C_3$ roots are $\pm {\epsilon }_i\pm {\epsilon }_j$ for $1\le i<j\le 3$ and $\pm 2{\epsilon }_i$ for $1\le i\le 3$. One root basis is ${\alpha }_1={\epsilon }_1-{\epsilon }_2$, ${\alpha }_2={\epsilon }_2-{\epsilon }_3$, ${\alpha }_3=2{\epsilon }_3$. With $A_{ij}=⟨{\alpha }_i,{\alpha }_j^{\vee }⟩$, its Cartan matrix isThe fundamental representation of highest weight $[1,0,0]$ is six-dimensional, with weights $\pm {\epsilon }_1,\pm {\epsilon }_2,\pm {\epsilon }_3$.

In the realization $\Phi (C_3)=\{\pm 2e_i,\pm e_i\pm e_j:i<j\}$, each coordinate two-plane cuts out a C2 root system. Each plane $\pm x_1\pm x_2\pm x_3=0$ cuts out an A2 root system; there are four such planes after identifying opposite normals. Thus the irreducible full rank-two root subsystems comprise three of type $C_2$ and four of type $A_2$. For an $A_2$ simple pair $p,q$, equal lengths and angle $120$ degrees give roots $\pm p,\pm q,\pm (p+q)$. For a mixed-length simple pair $s,l$ with $\Vert l{\Vert }^2=2\Vert s{\Vert }^2$ and $s\cdot l=-\Vert s{\Vert }^2$, the B2 root system has roots $\pm s,\pm l,\pm (s+l),\pm (2s+l)$. These lists also verify that the selected pairs are simple, rather than simply independent.

The C3 root system consists of $\pm 2e_i$ and $\pm e_i\pm e_j$ in ${\mathbb{R}}^3$. Its Weyl group is the signed symmetric group on three coordinates, of order $48$. The reflecting planes are $x_i=0$ and $x_i=\pm x_j$. The open Weyl chamber $x_1>x_2>x_3>0$ has closure generated by $(1,0,0)$, $(1,1,0)$ and $(1,1,1)$. Sign choices and absolute-coordinate orderings give its $48$ translates. On the unit sphere the chamber is a spherical triangle with angles $\pi /2$, $\pi /3$, $\pi /4$, and hence area $\pi /12$; the areas of all $48$ triangles sum to $4\pi$.

For the C3 root system, take ${\alpha }_1=e_1-e_2$, ${\alpha }_2=e_2-e_3$, ${\alpha }_3=2e_3$. With columns indexing coroots, the Cartan matrix isThe convention with rows indexing coroots gives $A^T$. Both describe a double bond between the short root ${\alpha }_2$ and long root ${\alpha }_3$, with arrow toward ${\alpha }_2$.