1.1.19 The DihedralGroup and Symmetry Transformations $\in 3\mathbb{C}$

We can con­sid­er the form of the $S_{3}/D_{3}$ sym­met­ric group from the last sec­tion and allow the per­mu­ta­tion ma­tri­ces to be $\in \mathbb{C}.$By doing this, the values for the cor­ners of the equi­lat­er­al tri­an­gle are not only $\in \mathbb{R}^{+}$, they take on the pos­si­bil­i­ty of being imag­i­nary valued. We define this higher-di­men­sion­al spa­tial free­dom by the fol­low­ing per­mu­ta­tion ma­tri­ces

for which, the com­po­si­tion of its func­tions is $\in \mathbb{C}^{\pm}.$By set­ting the equi­lat­er­al tri­an­gle into a higher-di­men­sion­al geom­e­try $\in \mathbb{C}^{3}$, means it not only ro­tates and re­flects $\in \mathbb{R}^{3}\ like\ S_{3}$, but is now free to move in the three or­thog­o­nal planes $\in \mathbb{C}^{3}$. These planes are in an or­thog­o­nal ori­en­ta­tion with one an­oth­er, form­ing com­pos­ite axes with values in $\in \mathbb{C}$ (Fig­ures 1.1.5-6).

To avoid con­fu­sion, we rep­re­sent the ver­tices of the equi­lat­er­al tri­an­gle by using the set $\left\{ A,B,C \right\}\in \mathbb{C}$, in­stead of using num­bers for its values. To present the iden­ti­ty el­e­ment $\epsilon$ in its geo­met­ric ori­en­ta­tion, the angles $A,B,C$ are ori­ent­ed along the re­spec­tive $X_{1},X_{2},X_{3}$ real-valued axes, as seen in Figure 1.1.7 below.

In the $S_{3}$ sym­me­try group, we had clock­wise and coun­ter­clock­wise ro­ta­tions, and re­flec­tions of the equi­lat­er­al tri­an­gle. This changes in $\mathbb{C}^{3}$ to an extent where the pre­vi­ous­ly seen re­flec­tion be­comes a tri­an­gle ro­ta­tion in com­plex space about an axis. To demon­strate this, we use the com­po­si­tion of the $"x"$per­mu­ta­tion func­tion on the iden­ti­ty $\epsilon$, $\phi(x):\epsilon$, acting on the vector $ABC$, with the de­fined mul­ti­pli­ca­tion op­er­a­tion

De­pict­ed in the figure below, we see that all vertex values are now imag­i­nary (ro­tat­ed), and $iB,iC$ have been re­flect­ed.

Con­tin­u­ing to ex­plore com­plex sym­me­tries, we take the com­pos­ite of per­mu­ta­tion func­tions

find­ing vertex co­or­di­nate values $\in \mathbb{R}^{-}$, de­pict­ed below in Figure 1.1.9.

It should be noted that the per­mu­ta­tions leave the cor­ners of the equi­lat­er­al tri­an­gle ori­ent­ed on the bound­aries of the com­plex cir­cles, rep­re­sent­ing the planes of the geo­met­ric group. The tri­an­gle stays intact as a reg­u­lar poly­gon for the per­mu­ta­tions of the group, lead­ing to the de­f­i­n­i­tion:

De­f­i­n­i­tion (1.1.93) – The Di­he­dral Group $\mathbf{D}_{\mathbf{3}\mathbb{C}}$ is a group under com­po­si­tion, sit­ting in the com­plex geo­met­ric space $\mathbb{C}^{3 \times 3}$, with a set of per­mu­ta­tion el­e­ments $A^{\mathbb{C}}$

The set of per­mu­ta­tion el­e­ments $A^{\mathbb{C}} \in D_{3\mathbb{C}}$ have values $\in \mathbb{C}^{\pm}$, many of which can be cal­cu­lat­ed with com­pos­ite func­tion arrange­ments of the two gen­er­a­tor el­e­ments, $X,Y$,

Our de­fined group $D_{3\mathbb{C}}$ has $12\ elements$, is non-abelian, and any per­mu­ta­tion $\phi$ is both an in­jec­tive and sur­jec­tive func­tion, there­fore bi­jec­tive, from the set of three vari­ables rep­re­sent­ing the sym­me­tries of the equi­lat­er­al tri­an­gle $\{ A,B,C\} \in \mathbb{C}^{3}$, onto itself.

The­o­rem (1.1.95) – The set of per­mu­ta­tions${\ A}^{\mathbb{C}} \in D_{3\mathbb{C}}$ forms a group under com­po­si­tion

Proof – Ver­i­fi­ca­tion of group axioms

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

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

• Iden­ti­ty: The iden­ti­ty func­tion $\epsilon_{A^{\mathbb{C}}}$ bi­jec­tive­ly maps all set el­e­ments${\ A}^{\mathbb{C}}$ 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.

$\square$

The reader may have no­ticed that do not refer to the $D_{3\mathbb{C}}$ group as being iso­mor­phic to $S_{3\mathbb{C}}$ at this point. The reason is that $D_{3\mathbb{C}}$ exists as a sub­group of $S_{3\mathbb{C}}$, having only twelve el­e­ments, $Y_{j} \in \mathbb{R}^{\pm}$ and $X_{j} \in \mathbb{C}^{\pm}$, where­as $S_{3\mathbb{C}}$ would have all the pos­si­ble el­e­ments for $A \in \mathbb{R}^{\pm}\ and\ \mathbb{C}^{\pm}.$This sec­tion has taken the di­he­dral/sym­me­try group, $D_{3}/S_{3}$, and ex­tend­ed its tri­an­gu­lar ro­ta­tions into the imag­i­nary-num­bered space. We have dou­bled the number of group per­mu­ta­tions by this geo­met­ric ex­ten­sion, while ad­her­ing to our group axioms. We see that the per­mu­ta­tions, $X \in A^{\mathbb{c}}$, are per­mu­ta­tions $\in Im^{\pm}$, and that, $Y \in A^{\mathbb{c}}$, are per­mu­ta­tions $\in \mathbb{R}^{\pm}$. The con­cept presents itself that some per­mu­ta­tions exist on the imag­i­nary axes while others may exist on the real axes.