1.1.16 The Dihedral Groups

Before we pro­ceed fur­ther, a clar­i­fi­ca­tion re­gard­ing the dis­tinc­tion be­tween sym­met­ric and di­he­dral groups is in order. Sym­met­ric groups rep­re­sent the entire com­bi­na­to­r­i­al set of per­mu­ta­tions pos­si­ble for the order of that group. For ex­am­ple, the sym­met­ric group $S_{n}$, has $n!$per­mu­ta­tions. The di­he­dral groups rep­re­sent the group of sym­me­tries of a reg­u­lar poly­gon, in­clud­ing the form's spe­cif­ic ro­ta­tions and re­flec­tions. Di­he­dral groups en­com­pass the rigid motion of poly­gons, which means the bi­jec­tion of a set of points onto itself, which leaves the dis­tance be­tween any two points on the poly­gon un­changed.

De­f­i­n­i­tion (1.1.80) – The Di­he­dral Group $\mathbf{D}_{\mathbf{n}}$ is the group of sym­me­tries of a reg­u­lar poly­gon, it is a group under com­po­si­tion, with a set of $2n$ per­mu­ta­tion el­e­ments $A$, that is bi­jec­tive, rep­re­sent­ing the sym­me­tries of a reg­u­lar poly­gon with $n$ ro­ta­tion el­e­ments and $n$ re­flec­tion el­e­ments

The $D_{3}$ group rep­re­sents the rigid ro­ta­tion­al and re­flec­tive sym­me­tries of the equi­lat­er­al tri­an­gle, and we will be work­ing with this group within var­i­ous geo­met­ric spaces through­out this book.

We choose to rep­re­sent the per­mu­ta­tion func­tions of $D_{3}$ as $n \times n$ square ma­tri­ces, which have one entry equal to 1 in each row and column, while the re­main­ing en­tries are 0. These ma­tri­ces are called per­mu­ta­tion ma­tri­ces, and we can rewrite the $D_{3}$ el­e­ments as

In Figure 1.1.3, we show the six per­mu­ta­tions of the equi­lat­er­al tri­an­gle rep­re­sent­ing the geo­met­ric re­flec­tions and ro­ta­tions of the $D_{3}$ group. It is straight­for­ward to see that we can per­mute the equi­lat­er­al tri­an­gle with matrix-matrix mul­ti­pli­ca­tion or matrix-vector mul­ti­pli­ca­tion. For ex­am­ple, the com­po­si­tion $\phi\left( y^{2} \right) \circ \phi(xy):\epsilon$, is

and in vector no­ta­tion, the above can be rep­re­sent­ed as

The com­po­si­tion of these six el­e­ments acts on the equi­lat­er­al tri­an­gle, rep­re­sent­ing all the geo­met­ric re­flec­tions and ro­ta­tions of the $D_{3}$ group. When first learn­ing about the $S_{3}/D_{3}$ group, stu­dents are shown per­mu­ta­tions of the equi­lat­er­al tri­an­gle as de­pict­ed in Figure 1.1.3, but the geo­met­ric space in which this poly­gon sits is not prop­er­ly con­veyed. To rotate and re­flect a form such as a poly­gon re­quires it to sit in the ap­pro­pri­ate geo­met­ric space where these can occur. It turns out that the equi­lat­er­al tri­an­gle in $S_{3}/D_{3}$, sits in a higher-di­men­sion­al hy­per­bol­ic space, de­pict­ed in Figure 1.1.4, where the pos­i­tive iden­ti­ty el­e­ment $\epsilon$ of the group is ori­ent­ed in the pos­i­tive quad­rant. The ro­ta­tions and re­flec­tions of the equi­lat­er­al tri­an­gle from the $S_{3}/D_{3}$ group will later be shown to rep­re­sent the tan­gent space of the $SO(3,3)$ Lie group.

The geo­met­ric space in which the re­flec­tions and ro­ta­tions of any form occur is an im­por­tant aspect of group theory. For the iden­ti­ty el­e­ment $\epsilon \in S_{3}/D_{3}$ in Figure 1.1.4 above, with all its el­e­ments $\in \mathbb{R}^{3 +}$ space, the ver­tices of the equi­lat­er­al tri­an­gle sit on the only shared co­or­di­nates be­tween or­thog­o­nal com­plex unit cir­cles $\in \mathbb{C}^{3}$, and the hy­per­bo­lae $\in \mathbb{R}^{3}$ space. This shared space is equiv­a­lent to set­ting $x_{j} = 0$, in the ex­po­nen­tials $e^{ix_{j}},e^{x_{j}}$, which places the ver­tices at points a unit length from the origin on the real axes only. These geo­met­ric space con­sid­er­a­tions will become more ap­par­ent as we study their Lie group rep­re­sen­ta­tions, where in­fin­i­tes­i­mal ro­ta­tions will orient the ver­tices beyond the re­stric­tions of the finite ro­ta­tions and re­flec­tions of the $S_{3}/D_{3}$ group.