1.1.26 The Symmetric/Dihedral Cone Group

De­f­i­n­i­tion (1.1.198) – We now define the Sym­met­ric/Di­he­dral group $\mathbf{D}_{\mathbf{3}\mathbb{c}}$, as a group under com­po­si­tion that sits in the com­plex geo­met­ric space $\mathbb{C}^{3 \times 3}$, having a set of per­mu­ta­tion el­e­ments $A^{\mathbb{c}}$

The set of per­mu­ta­tion el­e­ments $X \in A^{\mathbb{c}}$ of $D_{3\mathbb{c}}$ have multi-valued en­tries $\in \mathbb{C}^{3 \times 3}$ for the $X_{j,k}$ per­mu­ta­tions, and values $\in \mathbb{C}^{3 \times 3}$ for the cyclic sub­group $Y_{j,k}$, seen below.

Our de­fined group $S_{3\mathbb{c}}/D_{3\mathbb{c}}$ has $24$ el­e­ments, is non-abelian, and any per­mu­ta­tion $\phi$ is both an in­jec­tive and sur­jec­tive (bi­jec­tive) func­tion from the set of three vari­ables, rep­re­sent­ing the equi­lat­er­al tri­an­gle sym­me­tries $\{ A,B,C\} \in \mathbb{C}^{3}$ onto itself. For our abelian cyclic group, we have el­e­ments that rotate to values $\pm \ i, \pm 1$ on the axes of the com­plex unit circle.

The­o­rem (1.1.205) – The set of per­mu­ta­tions $A^{\mathbb{c}} \in D_{3\mathbb{c}}$ forms a group under com­po­si­tion

Proof – Ver­i­fi­ca­tion of group axioms

• Clo­sure: All el­e­ments $\in A^{\mathbb{c}}$ are bi­jec­tive, and the com­po­si­tion of any el­e­ments $\phi \circ \psi\ \in A^{\mathbb{c}}$is also bi­jec­tive.

• As­so­cia­tiv­i­ty: Per­mu­ta­tions are func­tions, and their com­po­si­tions are as­so­cia­tive.

• Iden­ti­ty: The iden­ti­ty func­tion $\epsilon_{A^{\mathbb{c}}}$ bi­jec­tive­ly maps all el­e­ments of the set $A^{\mathbb{c}}$ to them­selves $\left( \epsilon_{A^{\mathbb{c}}}:A^{\mathbb{c}} \rightarrow A^{\mathbb{c}} \right).$

• In­verse: $\phi$ is a per­mu­ta­tion and thus bi­jec­tive, and a func­tion exists$\ \phi^{- 1}$ that is also bi­jec­tive.

$\square$

Much like the Figure 1.1.5 plot of the $S_{3}$ iden­ti­ty in hy­per­bol­ic space, the iden­ti­ty el­e­ment $\epsilon(ABC)$ has its ver­tices aligned along the real axes, shown in Figure 1.1.17.

Taking the prod­uct $X_{1,1}$ with the iden­ti­ty el­e­ment $\epsilon = Y_{1,1}$, yields the com­pos­ite func­tion $\phi\left( X_{1,1} \right):\epsilon$, which re­flects the ver­tices $B,C$ along the axis $A$, and ro­tates the equi­lat­er­al tri­an­gle in all three of the com­plex $X_{j}$-planes to the co­or­di­nates $ACB = \frac{(1 + i)}{\sqrt{2}}(1,1,1)$ (Figure 1.1.18). We can con­tin­ue by taking the com­pos­ite func­tion from above and the $Y_{2,2}$ per­mu­ta­tion, to see a cyclic order shift $ACB \rightarrow BAC$ occurs for the tri­an­gle, having ro­tat­ed to the ad­ja­cent quad­rant values $BAC = \frac{(i - 1)}{\sqrt{2}}(1,1,1)$, seen in Figure 1.1.19 below.

In gen­er­al, the com­po­si­tion of func­tions iso­lat­ed to the $Y_{j,k}$ sub­group will result in ro­ta­tions and re­flec­tions of the equi­lat­er­al tri­an­gle that lay along the imag­i­nary and real axes of the $\mathbb{C}^{3 \times 3}$ space. Com­pare this to the com­po­si­tion of per­mu­ta­tion func­tions from the $X_{j,k}$ sub­group, which will rotate and re­flect the ver­tices of the equi­lat­er­al tri­an­gle to values which are $n = \left\{ 0, \pm 1, \pm 2, \pm 3,\ldots \right\},$ $\frac{(\pi + 2\pi n)}{4}$ ra­di­ans away from any real and imag­i­nary-axis in the geo­met­ric space.

This sec­tion has ex­pand­ed our di­he­dral group's de­f­i­n­i­tion to in­clude com­plex-valued per­mu­ta­tions with el­e­ments that per­fect­ly bal­ance real and imag­i­nary values. The $D_{3\mathbb{c}}$ group has Lie al­ge­bra el­e­ments com­posed of an even bal­ance of real and imag­i­nary num­bers, which will be cen­tral to the study of light in the up­com­ing physics volume.