1.4.8 The $\mathcal{Y}$ Subgroup of the Symmetric Conic Group

We have pre­vi­ous­ly men­tioned that the groups $D_{3\mathbb{C}}^{\boxplus}$ and $D_{3\mathbb{C}}^{\Delta}$ share the same $\mathcal{Y}$-sub­group, having el­e­ments of the form,

and axis num­bered de­ter­mi­nants. There is a con­spic­u­ous ab­sence of the el­e­ments $\in \mathcal{Y}$, having cone num­bered de­ter­mi­nant values, which define the $D_{3\mathbb{C}}^{\boxplus},D_{3\mathbb{C}}^{\Delta}$ groups. These el­e­ments were deemed nec­es­sary to com­plete the $S_{3\mathbb{C}}^{\Delta}$ sym­met­ric conic group. The $\mathcal{Y}$-sub­group $\subset$ $S_{3\mathbb{C}}^{\Delta}$, con­tains all the pos­si­ble com­bi­na­tions of axis and cone num­bers, and in ad­di­tion to the stated axis de­ter­mi­nant forms, the in­clu­sion of the cone de­ter­mi­nant el­e­ments round out the sub­group,

For the $\mathcal{Y}$-el­e­ments, eigen­val­ues will take the form

and solv­ing for the char­ac­ter­is­tic poly­no­mi­als of $\mathcal{Y}_{j}$, we find $\mathcal{Y}_{1}$ first

for $\mathcal{Y}_{2}$

whose terms $a,b,c$ can take on cone number or axis number values, and ul­ti­mate­ly, we find a sim­i­lar result for the char­ac­ter­is­tic poly­no­mi­al of $\mathcal{Y}_{3}$

Since the $a,b,c$ en­tries in each matrix can be either a cone or an axis number, $\mathcal{Y}_{j}$ can have the fol­low­ing pos­si­ble eigen­val­ues

c. $\left( 1,( - 1)^{\frac{2}{3}},{- ( - 1)}^{\frac{1}{3}} \right)_{\lambda^{3} = 1}$, d. $\left( - i,( - 1)^{\frac{1}{6}},( - 1)^{\frac{5}{6}} \right)_{\lambda^{3} = i}$.

c. $\left( ( - 1)^{\frac{5}{12}},{- ( - 1)}^{\frac{1}{12}},{- ( - 1)}^{\frac{3}{4}} \right)_{\lambda^{3} = {- ( - 1)}^{\frac{1}{4}}}$, d. $\left( ( - 1)^{\frac{7}{12}}, - ( - 1)^{\frac{1}{4}}, - ( - 1)^{\frac{11}{12}} \right)_{\lambda^{3} = - ( - 1)^{\frac{3}{4}}}$.

Com­par­ing pos­si­ble eigen­val­ues for $\mathcal{Y}_{2},\mathcal{Y}_{3}$, we find el­e­ments with axis number de­ter­mi­nants, having four pos­si­ble values (Figure 1.4.16)

and $\mathcal{Y}_{2},\mathcal{Y}_{3}$, having pos­si­ble cone number de­ter­mi­nants from the set of the four fol­low­ing values (Figure 1.4.17)

When con­sid­er­ing pos­si­ble eigen­val­ues of $\mathcal{Y}_{1}$, the $a,b,c$, el­e­ments are on the main di­ag­o­nal, which leads to eigen­val­ue so­lu­tions that equal to each entry on matrix di­ag­o­nal

mean­ing, we can have mixed values in the form of cone num­bers and axis num­bers. This is quite dif­fer­ent than the $\mathcal{Y}_{2},\mathcal{Y}_{3}$ eigen­val­ues, which were bound to one number or the other. This gives the $\mathcal{Y}_{1}$ el­e­ments a more di­verse abil­i­ty to scale el­e­ments in the sym­met­ric conic group.