$\displaystyle\Phi_{\mathrm{t}}^{+}(p_{3})=\frac{2\mu}{3\sqrt{3}},$ (203)
$\displaystyle\Phi_{\mathrm{t}}^{+}(p_{4})$ $\displaystyle=$
$\displaystyle\Phi_{\mathrm{t}}^{+}(p_{5})=\mathrm{sgn}(\mu)\frac{4}{3\sqrt{3}\sqrt{4-\mu^{2}}}.$
(204)

These critical values are plotted in figure 24 as functions of $\mu$.

Figure 24: The critical values of the potential $\Phi_{\mathrm{t}}^{+}$ as a
functions of $\mu\in[-\sqrt{2},\sqrt{2}]$. In black,
$\Phi_{\mathrm{t}}^{+}(p_{1})=0$; in blue,
$\Phi_{\mathrm{t}}^{+}(p_{2})=\Phi_{\mathrm{t}}^{+}(p_{3})$; in red,
$\Phi_{\mathrm{t}}^{+}(p_{4})=\Phi_{\mathrm{t}}^{+}(p_{5})$.

######  Remark 16.

Note that $\Phi_{\mathrm{t}}^{+}(p_{2})=\Phi_{\mathrm{t}}^{+}(p_{3})$, which
corresponds to $\Phi_{\mathrm{t}}^{+}$ being even in $x_{2}$ on the $x_{1}=0$
line; and $\Phi_{\mathrm{t}}^{+}(p_{4})=\Phi_{\mathrm{t}}^{+}(p_{5})$, which
corresponds to the invariance of $\Phi_{\mathrm{t}}^{+}$ under inversion in
$x_{1}$.

####  5.2.7 Nature of critical points.

Having identified the critical points, we should enquire if these are maxima,
minima, or saddles. In order to ascertain the nature of the critical points,
we should consider the matrix of second derivatives

$\mathsf{H}:=\nabla^{2}\Phi_{\mathrm{t}}^{+}=\pmatrix{\frac{\partial^{2}\Phi_{\mathrm{t}}^{+}}{\partial
x_{1}^{2}}&\frac{\partial^{2}\Phi_{\mathrm{t}}^{+}}{\partial x_{1}\partial
x_{2}}\cr\frac{\partial^{2}\Phi_{\mathrm{t}}^{+}}{\partial x_{1}\partial
x_{2}}&\frac{\partial^{2}\Phi_{\mathrm{t}}^{+}}{\partial x_{2}^{2}}\cr},$
(205)

and compute its eigenvalues, or at least their sign, at the critical points
$p_{j}$ identified in section 5.2.6 above.

We will stick to our assumption that $A_{1}=0$ and $A_{2}=+1$ (with
$A_{3}=\mu$); the case of negative $A_{2}$ can be recovered recalling that a
change of sign in $A_{2}$ corresponds to a change of sign in the potential,
and that $\mu=A_{3}/A_{2}$ (so a change of sign in $A_{2}$ leaving $A_{3}$
unchanged corresponds to a change of sign in $\mu$, while a change of sign in
both $A_{2}$ and $A_{3}$ leaves $\mu$ unchanged).

For the first three critical points the eigenvalues are easily computed and
provide simple formulas:

$\displaystyle\cases{p_{1}:\lambda_{1}=0,\quad\lambda_{2}=2\mu,\\\
p_{2}:\lambda_{1}=-4\sqrt{3}\mu,\quad\lambda_{2}=-\sqrt{2/3}\left(2+\sqrt{2}\mu\right),\\\
p_{3}:\lambda_{1}=-4\sqrt{3}\mu,\quad\lambda_{2}=+\sqrt{2/3}\left(2-\sqrt{2}\mu\right).}$
(206)

Note that in the degenerate case where $\mu=0$ the first eigenvalue of all
these critical point vanishes, while for $\mu=\pm\sqrt{2}$ the second
eigenvalue vanishes in one.

######  Remark 17.

This simple analysis is not conclusive for the critical point at the North
pole. Actually, the series expansion along $x_{2}=0$ is flat to all orders, as
clear from the explicit expression of $\Phi_{\mathrm{t}}$.

The stability of points $p_{2}$ and $p_{3}$ is promptly analyzed:

  * •

The point $p_{2}$ is a minimum for $\mu>0$, a saddle for $-\sqrt{2}<\mu<0$,
and a maximum for $\mu<-\sqrt{2}$;

  * •

The point $p_{3}$ is a minimum for $\mu<0$, a saddle for $0<\mu<\sqrt{2}$, and
a maximum for $\mu>\sqrt{2}$.

For the other critical points, $p_{4}$ and $p_{5}$, we find it more convenient
to characterize their nature by computing the trace and determinant of the
Hessian matrix $\mathsf{H}$ in (205) in terms of the parameter $\xi$
introduced in (200),

$\tr\mathsf{H}=\frac{2\sqrt{6}(4\cos^{4}\xi-11\cos^{2}\xi+12)}{3\sin\xi(2\cos^{2}\xi-1)},\quad\det\mathsf{H}=\frac{16\cos^{2}\xi}{1-2\cos^{2}\xi},$
(207)

where use has also been made of the inverse of the function in (200),

$\mu=\frac{\sqrt{2}}{\sin\xi}\sqrt{2\sin^{2}\xi-1}.$ (208)

It is a simple matter to conclude from the study of the signs of
$\tr\mathsf{H}$ and $\det\mathsf{H}$ that

  * •

For $-\sqrt{2}<\mu<0$ the points $p_{4}$ and $p_{5}$ are local minima;

  * •

For $0<\mu<\sqrt{2}$ the points $p_{4}$ and $p_{5}$ are local maxima.

These results can be confirmed by computing numerically the index for the
different critical points; such computations are summarized in table 2,
showing the index of critical points $p_{2}$-$p_{5}$ for different intervals
of values of $\mu$ (recall that $p_{4}$ and $p_{5}$ only exist for
$0<|\mu|\leq\sqrt{2}$).

Table 2: Critical points of the potential $\Phi_{\mathrm{t}}$ in (180) for
$A_{2}=1$ and different values of $\mu$. For $|\mu|>\sqrt{2}$, neither $p_{4}$
nor $p_{5}$ exists.

  * point | $\mu<-\sqrt{2}\ $ | $-\sqrt{2}<\mu<0\ $ | $0<\mu<\sqrt{2}\ $ | $\sqrt{2}<\mu$
---|---|---|---|---
$p_{2}$ | $1$ | $-1$ | $1$ | 1
$p_{3}$ | $1$ | $1$ | $-1$ | 1
$p_{4}$ | — | $1$ | $1$ | —
$p_{5}$ | — | $1$ | $1$ | —

######  Remark 18.

As for the degenerate critical point $p_{1}$, we have not computed directly
its index, but the Poincaré-Hopf theorem requires $p_{1}$ to be a saddle, as
the total index of all critical points must be $\iota=+2$ on the whole sphere
$\mathbb{S}^{2}$.

In figure 25 we present the contour plots of $\Phi_{\mathrm{t}}^{+}$ for
different values of $\mu$. They are accompanied in figure 26 by the
corresponding polar plots.

(a) $\mu=-2$

(b) $\mu=-\sqrt{2}$

(c) $\mu=-1/2$

(d) $\mu=-1$

(e) $\mu=1/2$

(f) $\mu=1$

(g) $\mu=2$

(h) $\mu=\sqrt{2}$

Figure 25: Contour plots of the potential $\Phi_{\mathrm{t}}^{+}$ for
$A_{3}=\mu A_{2}$ and $A_{2}=1$ on the $(x_{1},x_{2})$ plane, for different
values of $\mu$.

(a) $\mu=-2$

(b) $\mu=-\sqrt{2}$

(c) $\mu=-1/2$

(d) $\mu=-1$

(e) $\mu=1/2$

(f) $\mu=1$

(g) $\mu=2$

(h) $\mu=\sqrt{2}$

Figure 26: Polar plots of the potential $\Phi_{\mathrm{t}}$ corresponding to
the contour plots in figure 25. Here the protruding lobes designate maxima
(and invaginated minima). The origin is $p_{1}$, a degenerate saddle for all
values of $\mu$, accompanied by a non-degenerate saddle lying on the
$x_{2}$-axis (either $p_{2}$ or $p_{3}$ in (197), depending on the sign of
$\mu$).

###  5.3 Full potential

We want now to consider a general potential $\Phi_{\mathrm{s}}$, i.e., the
superposition of a traceless potential $\Phi$, see equation (159), and of a
trace type potential $\Phi_{\mathrm{t}}$, see equation (160).

The rich phenomenology displayed by the traceless part $\Phi$ can only be
enriched by considering also a trace type part; a complete analysis would most
likely lead to a rather complicate discussion. Here our study will be confined
to a simple, explanatory case.

The most striking feature arising from the analysis of traceless tensor
potentials is maybe the presence of an exactly tetrahedral phase [44]; we
wonder if such a phase can also exist in the presence of a pure trace type
contribution. The general traceless potential $\Phi$ is written as in (159).
The situation in which it enjoys full tetrahedral symmetry is obtained in
section 7.2 of [44] for

$\alpha_{0}=0,\ \alpha_{1}=0,\ \alpha_{2}=\pm\frac{1}{\sqrt{2}},\
\alpha_{3}=1,\ \beta_{3}=-\frac{1}{2}.$ (209)

In this way the oriented traceless potential $\Phi$ reads as

$\Phi_{\mathrm{T}}:=x_{3}^{3}-\frac{3}{2}\left(x_{1}^{2}+x_{2}^{2}\right)x_{3}+\frac{1}{\sqrt{2}}\left(x_{2}^{2}-3x_{1}^{2}\right)x_{2},$
(210)

where we have set $\alpha_{2}=1/\sqrt{2}$, for definiteness.

The four maxima of the potential $\Phi_{\mathrm{T}}$ are located at the
vertices of a regular tetrahedron, and more specifically at the points given
in three-dimensional Cartesian coordinates by (see [44])

$\quad\cases{p_{1}=(0,0,1),\\\
p_{2}=\left(0,\frac{2\sqrt{2}}{3},-\frac{1}{3}\right),\
p_{3}=\left(-\sqrt{\frac{2}{3}},-\frac{\sqrt{2}}{3},-\frac{1}{3}\right),\
p_{4}=\left(\sqrt{\frac{2}{3}},-\frac{\sqrt{2}}{3},-\frac{1}{3}\right).}$
(211)

####  5.3.1 Perturbation approach.

Here we only consider an extreme case, i.e., that in which one of the two
parts (the traceless one) can be considered as dominant, and the other one
(the pure trace one) as a perturbation.

It should be stressed that we cannot arbitrarily orient both the traceless and
the trace type part of the potential at the same time: we can only orient one
of these (or their sum). We find it more convenient to orient the traceless
part, in particular when considering the trace type part as a perturbation.

As mentioned above, we want to investigate if a potential $\Phi_{\mathrm{s}}$
including both the traceless and the pure trace parts can have maxima at the
same critical points (211), i.e., display a tetrahedral symmetry for the
physical states (identified by maxima; see [44]. It should be noted that we
only require the locations of the maxima to be mapped unto one another by the
action of the tetrahedral group $T_{d}$: we are not requiring, in general,
that the values of these maxima are the same (as was the case in the
tetrahedral potential). Thus, letting $\Phi_{\mathrm{T}}$ be as in (210), we
shall write $\Phi_{\mathrm{s}}=\Phi_{\mathrm{T}}+\Phi_{\mathrm{t}}$ as in
(158) with coefficients given by

$\quad\cases{\alpha_{0}=\varepsilon(\delta\alpha_{0}),\
\alpha_{1}=\varepsilon(\delta\alpha_{1}),\
\alpha_{2}=\frac{1}{\sqrt{2}}+\varepsilon(\delta\alpha_{2}),\ \
\alpha_{3}=1+\varepsilon(\delta\alpha_{3}),&\\\
\beta_{1}=\varepsilon(\delta\beta_{1}),\
\beta_{2}=\varepsilon(\delta\beta_{2}),\
\beta_{3}=-\frac{1}{2}+\varepsilon(\delta\beta_{3}),&\\\
A_{1}=\varepsilon(\delta A_{1}),\ A_{2}=\varepsilon(\delta A_{2}),\
A_{3}=\varepsilon(\delta A_{3}).&}$ (212)

Here $\varepsilon$ is a small parameter, all the other newly introduced
parameters are expected to be of order one.131313In general, the prescription
$\alpha_{1}=\beta_{1}=\beta_{2}=0$, which ensures the orientation of the
traceless potential, can be violated.

We then look for critical points of $\Phi_{\mathrm{s}}$ at first order in
$\varepsilon$, and require the points $p_{1}$ \- $p_{4}$ in (211) to be still
critical points (and, by a perturbation argument, hence necessarily maxima)
for $\Phi_{\mathrm{s}}$. Through some standard algebra, we find that this is
the case, provided that

$\displaystyle\quad\cases{\delta\alpha_{0}=\frac{\sqrt{2}}{3}\delta\alpha_{1},\\\
\delta\beta_{1}=\frac{1}{3}\delta\alpha_{1},\ \delta\beta_{2}=0,\\\ \delta
A_{1}=4\delta\alpha_{1},\ \delta
A_{2}=2\delta\alpha_{2}+\frac{1}{\sqrt{2}}\delta\alpha_{3}+3\sqrt{2}\delta\beta_{3},\
\delta
A_{3}=-\sqrt{2}\delta\alpha_{2}+\frac{5}{2}\delta\alpha_{3}+3\delta\beta_{3},}$
(213)

where $\delta\alpha_{1}$, $\delta\alpha_{2}$, $\delta\alpha_{3}$, and
$\delta\beta_{3}$ are free parameters. We can afford being a bit more
restrictive and consider only perturbations of the traceless part that
preserve the orientation of this latter by keeping the constraints

$\alpha_{1}=\beta_{1}=\beta_{2}=0.$ (214)

We thus arrive at the conditions

$\cases{\delta A_{1}=0,&\\\ \delta
A_{2}=2\delta\alpha_{2}+\frac{1}{\sqrt{2}}\delta\alpha_{3}\
+3\sqrt{2}\,\delta\beta_{3},&\\\ \delta
A_{3}=-\sqrt{2}\delta\alpha_{2}+\frac{5}{2}\delta\alpha_{3}+3\delta\beta_{3},&}$
(215)

where $\delta\alpha_{2}$ , $\delta\alpha_{3}$, and $\delta\beta_{3}$ are
arbitrary constants.

######  Remark 19.

We have only required _maxima_ to remain at the same points. If we extend this
requirement to all critical points, it turns out that $\Phi_{\mathrm{s}}$ must
be proportional to $\Phi_{\mathrm{T}}$, thus neutralizing any contribution
from a pure trace tensor.

Making use of (215), (214), and (212) in (158), we can easily express the
symmetric potential $\Phi_{\mathrm{s}}=\Phi_{\mathrm{T}}+\Phi_{\mathrm{t}}$ in
the form

$\eqalign{\Phi_{\mathrm{s}}&=\frac{1}{\sqrt{2}}\
x_{2}\left(x_{2}^{2}-3x_{1}^{2}\right)-\frac{3}{2}\left(x_{1}^{2}-x_{2}^{2}\right)x_{3}+x_{3}\left(x_{3}^{2}-3x_{2}^{2}\right)\\\
&+\varepsilon\left[\left(2\delta\alpha_{2}+\frac{1}{\sqrt{2}}\delta\alpha_{3}+3\sqrt{2}\delta\beta_{3}\right)x_{2}x_{1}^{2}+\delta\alpha_{2}x_{2}\left(x_{2}^{2}-3x_{1}^{2}\right)\right.\\\
&\quad\left.+\left(-\sqrt{2}\delta\alpha_{2}+\frac{5}{2}\delta\alpha_{3}+3\delta\beta_{3}\right)x_{2}^{2}x_{3}+3\delta\beta_{3}\left(x_{1}^{2}-x_{2}^{2}\right)x_{3}+\delta\alpha_{3}x_{3}\left(x_{3}^{2}-3x_{2}^{2}\right)\right],}$
(216)

which is _not_ equivalent to $\Phi_{\mathrm{T}}$. In cases where the
(observable) physics is only described by the _location_ of the maxima (or the
minima) of the octupolar potential, we thus have that a perturbation of the
tetrahedral potential $\Phi_{\mathrm{T}}$ by a combination of the potentials
associated to a traceless and to a trace type tensors can still describe the
same physics.

######  Remark 20.

Physics could also depend on the (relative or absolute) levels of the maxima
of the octupolar potential in (216); thus it matters if they are at the same
level or not. A tedious, but easy calculation shows that requiring all maxima
of $\Phi_{\mathrm{s}}$ in (216) to be equal reduces $\Phi_{\mathrm{s}}$ to a
multiple of $\Phi_{\mathrm{T}}$. In other words, the only way to have
degenerate maxima at tetrahedral points is with a pure tetrahedral potential.

######  Remark 21.

We might ask for a smaller degeneration, i.e., require that the potential
$\Phi_{\mathrm{s}}$ in (216) takes the same value at the points $p_{2}$,
$p_{3}$, and $p_{4}$ in (211), albeit this is allowed to be different from the
value taken at the point $p_{1}$. In this case we have to require

$\delta\beta_{3}=-\frac{1}{\sqrt{6}}\left(2\sqrt{2}\delta\alpha_{2}+\delta\alpha_{3}\right).$
(217)

With this prescription, we get

$\Phi_{\mathrm{s}}(p_{1})=1+\varepsilon\delta\alpha_{3},\quad\Phi_{\mathrm{s}}(p_{2})=\Phi_{\mathrm{s}}(p_{3})=\Phi_{\mathrm{s}}(p_{4})=1+\varepsilon\left(\frac{8}{9}\sqrt{2}\delta\alpha_{2}+\frac{1}{9}\delta\alpha_{3}\right).$
(218)

We can further set $\delta\alpha_{3}=0$, so that the value of the potential at
the orienting maximum (in the North pole) is unchanged; in this case, setting
also $\delta\alpha_{2}=1$, we get

$\Phi_{\mathrm{s}}(p_{1})=1,\quad\Phi_{\mathrm{s}}(p_{2})=\Phi_{\mathrm{s}}(p_{3})=\Phi_{\mathrm{s}}(p_{4})=1+\varepsilon\frac{8}{9}\sqrt{2},$
(219)

and, more generally,

$\eqalign{\Phi_{\mathrm{s}}=\Phi_{\mathrm{T}}+\varepsilon\Bigg{[}&2\sqrt{2}x_{2}^{2}\sqrt{1-x_{1}^{2}-x_{2}^{2}}+\left(x_{2}^{2}-3x_{1}^{2}\right)x_{2}\\\
&+\sqrt{2}\left(x_{1}^{2}-x_{2}^{2}\right)\sqrt{1-x_{1}^{2}-x_{2}^{2}}\Bigg{]}.}$
(220)

####  5.3.2 Non-perturbation approach.

We can also proceed non-perturbatively. To this end, we consider a general
superposition $\Phi_{\mathrm{s}}$ of a traceless potential $\Phi$ as in (159)
and a trace type potential $\Phi_{\mathrm{t}}$ as in (160).

We then consider the gradient of $\Phi_{\mathrm{s}}$, evaluate it at the
points $p_{1}$-$p_{4}$ in (211), and require it vanishes there. Through
standard computations we obtain that this is the case provided some relations
hold between the different parameters characterizing the potential. These are
as follows141414The similarity between the last two equations in (221) and
(215) should be heeded.

$\cases{\alpha_{0}=\sqrt{2}\beta_{1},\quad\alpha_{1}=3\beta_{1},\quad\beta_{2}=0,&\\\
A_{1}=12\beta_{1},&\\\
A_{2}=2\alpha_{2}+\frac{1}{\sqrt{2}}\alpha_{3}+3\sqrt{2}\beta_{3},&\\\
A_{3}=-\sqrt{2}\alpha_{2}+\frac{5}{2}\alpha_{3}+3\sqrt{2}\beta_{3}.&}$ (221)

Here $\alpha_{2}$, $\alpha_{3}$, $\beta_{1}$, and $\beta_{3}$ are free
parameters.

######  Remark 22.

Equations (221) guarantee that the tetrahedral points are critical. If we also
require that $p_{2}$-$p_{4}$ in (211) are all on the same level set of
$\Phi_{\mathrm{s}}$, we are led to the equations

$\alpha_{0}=\alpha_{3}=\beta_{1}=0,\quad\beta_{3}=-\frac{1}{6}(2\sqrt{2}\alpha_{2}+\alpha_{3}),$
(222)

which still leave $\alpha_{2}$ and $\alpha_{3}$ as free parameters. By (221),
the coefficients of the pure trace part $\Phi_{\mathrm{t}}$ then
become151515Note that we are in the degenerate case $A_{2}=0$.

$A_{1}=0,\quad A_{2}=0,\quad
A_{3}=2\left(\alpha_{3}-\sqrt{2}\alpha_{2}\right).$ (223)

By use of both (222) and (223), we give $\Phi_{\mathrm{s}}$ the special form

$\qquad\quad\Phi_{\mathrm{s}}=\frac{1}{2}\alpha_{3}x_{3}\left(2x_{3}^{2}-x_{1}^{2}-x_{2}^{2}\right)-\alpha_{2}\left[\left(3x_{2}+\sqrt{2}x_{3}\right)x_{1}^{2}+x_{2}^{2}\left(\sqrt{2}x_{3}-x_{2}\right)\right].$
(224)

######  Remark 23.

If now we also require that $\Phi_{\mathrm{s}}$ in (224) has in $p_{1}$ the
same value as in $p_{2}$-$p_{4}$, we easily see that it must be

$\alpha_{3}=\sqrt{2}\alpha_{2},$ (225)

which, by (223), makes $A_{3}$ vanish as well, so that $\Phi_{\mathrm{s}}$ is
eventually proportional (through $\alpha_{3}$) to $\Phi_{\mathrm{T}}$.

######  Remark 24.

Consider now the potential $\Phi_{\mathrm{s}}$ in (224), _without_ assuming
(225). We have seen that it has critical points at the tetrahedral points
(211). It should be noted that while working perturbatively we were guaranteed
the critical points at $p_{1}$-$p_{4}$ were, for $\varepsilon$ sufficiently
small, still maxima, in the present case this is not guaranteed. To this end,
we compute as in section 5.2.7 the eigenvalues of the Hessian matrix
$\mathsf{H}$ of $\Phi_{\mathrm{s}}$ in (224):

$\cases{p_{1}:\lambda_{1}=\lambda_{2}=-2\left(\sqrt{2}\alpha_{2}+2\alpha_{3}\right);\\\
p_{2}\textit{-}p_{4}:\lambda_{1}=-6\sqrt{2}\alpha_{2},\quad\lambda_{2}=-6\left(5\sqrt{2}\alpha_{2}+4\alpha_{3}\right).}$
(226)

These are all real, and we want all of them to be negative. It is easily seen
that this is the case, provided that

$\alpha_{2}>0,\quad\alpha_{3}>-\frac{\alpha_{2}}{\sqrt{2}}.$ (227)

Correspondingly, we obtain that $\Phi_{\mathrm{s}}$ in (224) satisfies

$\Phi_{\mathrm{s}}(p_{1})=\alpha_{3},\quad\Phi_{\mathrm{s}}(p_{2})=\Phi_{\mathrm{s}}(p_{3})=\Phi_{\mathrm{s}}(p_{4})=\frac{1}{9}\left(8\sqrt{2}\alpha_{2}+\alpha_{3}\right).$
(228)

So the three degenerate maxima in the Southern hemisphere are higher than the
maximum at the North pole if

$\alpha_{3}<\sqrt{2}\alpha_{2},$ (229)

and lower than the maximum at the North pole if in (229) the inequality is
reversed.

####  5.3.3 Invariance of the combined potential.

To illustrate the subtleties that may be hidden in the fully symmetric
potential $\Phi_{\mathrm{s}}$, we consider here an invariance property that
determines uniquely the traceless potential $\Phi$, but fails to determine the
combined potential resulting from adding to it a traceless component
$\Phi_{\mathrm{t}}$. In [45], we studied the action of the tetrahedron group
$T_{d}$ on traceless type potentials $\Phi$. Here, we discuss the action of
the same on the particular potential $\Phi_{\mathrm{s}}$ given by (224).

We will use the notation of [45], in particular the representation of $T_{d}$
as a group of matrices acting in $\mathbb{R}^{3}$. It was shown there that the
maximal subgroup $G$ of $T_{d}$ leaving the North pole fixed is made of the
matrices $\\{M_{1},M_{2},M_{3},M_{13},M_{14},M_{15}\\}$, in the notation
adopted there. Here we will rename these as $M_{1}$-$M_{6}$, which read
explicitly as

$\displaystyle\qquad\cases{M_{1}=\pmatrix{1&0&0\cr 0&1&0\cr
0&0&1},&$M_{2}=\pmatrix{-1/2&\sqrt{3}/2&0\cr-\sqrt{3}/2&-1/2&0\cr 0&0&1}$,\\\
M_{3}=\pmatrix{-1/2&-\sqrt{3}/2&0\cr\sqrt{3}/2&-1/2&0\cr
0&0&1},&$M_{4}=\pmatrix{-1&0&0\cr 0&1&0\cr 0&0&1}$,\\\
M_{5}=\pmatrix{1/2&-\sqrt{3}/2&0\cr-\sqrt{3}/2&-1/2&0\cr
0&0&1},&$M_{6}=\pmatrix{1/2&\sqrt{3}/2&0\cr\sqrt{3}/2&-1/2&0\cr 0&0&1}$.}$
(230)

The commutation relation among these can be read from [45]; the only
nontrivial subgroup is $G_{0}=\\{M_{1},M_{2},M_{3}\\}$.

It is a simple matter to check that for $\Phi_{\mathrm{s}}$ in (224),

$\Phi_{\mathrm{s}}(M\bm{x})=\Phi_{\mathrm{s}}(\bm{x})\qquad\forall M\in G.$
(231)

That is, $\Phi_{\mathrm{s}}$ is $G$-invariant. One might wonder if the
converse is also true, i.e., if the requirement of being $G$-invariant does
uniquely select $\Phi_{\mathrm{s}}$ in (224).

To discuss this matter, we start from the general expression for
$\Phi_{\mathrm{s}}$ in (158) and require

$\Phi_{\mathrm{s}}(M_{i}\bm{x})=\Phi_{\mathrm{s}}(\bm{x}),$ (232)

for $i=1,\dots,6$. Some elementary algebra shows that this amounts to
enforcing the conditions

$\cases{\alpha_{0}=0,\quad\alpha_{1}=0,\quad\beta_{1}=0,\quad\beta_{2}=0,\\\
A_{1}=0,\quad A_{2}=0,\quad A_{3}=3(\alpha_{3}+2\beta_{3}).}$ (233)

However, by direct computation we see that this choice of parameters does
_not_ make $\Phi_{\mathrm{s}}$ in (158) agree with (224), unless we set

$\beta_{3}=-\frac{1}{6}\left(2\sqrt{2}\alpha_{2}+\alpha_{3}\right),$ (234)

which, incidentally, implies the third of (223). We thus conclude that the
condition of $G$-invariance does _not_ determine uniquely $\Phi_{\mathrm{s}}$.

######  Remark 25.

The $G$-invariance condition determines uniquely the traceless potential
$\Phi$, whereas the trace potential $\Phi_{\mathrm{t}}$ is determined up to a
multiplicative factor $A_{3}$.

######  Remark 26.

It could also be mentioned that the degeneration of values at the critical
points $p_{2}$-$p_{4}$ does not only apply to the whole potential
$\Phi_{\mathrm{s}}$, but also to its traceless and trace components
separately, although these are not separately invariant.

##  6 Other Approaches

So far we have privileged descriptions of the properties of an octupolar
tensor $\mathbf{A}$ based upon the octupolar potential $\Phi$ introduced in
(20) and the several variants encountered above. Other approaches to these
properties have been proposed in the literature. We devote this section to
some of these, trying to establish connections with ours.

###  6.1 Maxwell multipoles

This approach to octupolar tensors in rooted in Maxwell’s multipole
representation of spherical harmonics [79, pp. 179–214] (see also [28, pp.
514–522]). Our account, phrased in a modern language, follows [131] (see also
[32] for a broader perspective). A theorem due to Sylvester [108] (alternative
proofs of which can also be found in [7] and [131]) put Maxwell’s method on a
solid mathematical ground.

There is a one-to-one correspondence between a completely symmetric tensor
$\bm{\mathsf{A}}\in\mathcal{T}(r,\mathsf{V})$ and a homogeneous polynomial
$P_{r}(\bm{x})$ of degree $r$ in $\bm{x}\in$ with $\dim\mathsf{V}=3$ as

$P_{r}(\bm{x})=\sum_{i_{1}i_{2}\dots i_{r}=1}^{3}A_{i_{1}i_{2}\dots
i_{r}}x_{i_{1}}x_{i_{2}}\dots x_{i_{r}}.$ (235)

Sylvester’s theorem says that, given a real homogeneous polynomial
$P_{r}(\bm{x})$ of degree $r\geqq 2$, there are $r$ vectors
$\bm{a}_{1},\bm{a}_{2},\dots,\bm{a}_{r}\in\mathsf{V}$ and a real homogeneous
polynomial $P_{r-2}(\bm{x})$ of degree $r-2$ such that

$P_{r}(\bm{x})=\prod_{s=1}^{r}(\bm{a}_{s}\cdot\bm{x})+(\bm{x}\cdot\bm{x})P_{r-2}(\bm{x}).$
(236)

Building on classical results, Zou and Zheng [131] proved that every tensor
$\bm{\mathsf{D}}^{(m)}$ in the decomposition of a fully symmetric tensor
$\bm{\mathsf{A}}$ in (28) can be represented as

$\bm{\mathsf{D}}^{(m)}=A_{m}\mbox{$\hskip
1.99997pt\stackrel{{\scriptstyle\mbox{$\vphantom{\rule[-2.58316pt]{0.51663pt}{3.0998pt}}\rule[-2.06653pt]{0.51663pt}{2.58316pt}\rule{41.65234pt}{0.51663pt}\rule[-2.06653pt]{0.51663pt}{2.58316pt}$}}}{{\mbox{$\displaystyle\bm{a}_{1}\otimes\bm{a}_{2}\cdots\bm{a}_{m}$}}}\hskip
1.99997pt$},$ (237)

where, for $m\leqq r$, $A_{m}>0$ is a scalar and
$\bm{a}_{1},\bm{a}_{2},\dots\bm{a}_{m}$ are vectors on the unit sphere
$\mathbb{S}^{2}$ in $\mathsf{V}$ determined uniquely by
$\bm{\mathsf{D}}^{(m)}$, to within a change of sign in an even number of them.

The poles designated on $\mathbb{S}^{2}$ by these vectors are called Maxwell’s
_multipoles_. The connection thus established between fully symmetric
traceless tensors and spherical harmonics justifies calling _harmonic_ these
tensors, as well as the decomposition in (26) for a generic tensor. This
connection is further explored in [4]. Harmonic tensors also play a role in
reconstructing the _crystallite orientation function_ for poly-crystalline
materials [93, 1, 50]; for this topic the reader is referred to the
comprehensive review of Man [75], in particular, to Chapt. 17.

######  Remark 27.

As shown in the early work of Backus [7], the multipole representation of
$\bm{\mathsf{A}}$, by its very geometric interpretation, can be effective in
identifying the symmetries of $\bm{\mathsf{A}}$. For more recent contributions
to the role played by harmonic decomposition of a tensor $\bm{\mathsf{A}}$ in
identifying all symmetry classes it may belong to, the reader is referred to
the works [8, 9, 42, 43, 5, 6]. This is, however, a slippery terrain, as
witnessed for example by the disagreement between [46] and [130], which for
the piezoelectric tensor (see section 2.4.3) found with different methods $14$
and $15$ symmetry classes, respectively.

When applied to $\mbox{$\hskip
1.99997pt\stackrel{{\scriptstyle\mbox{$\vphantom{\rule[-2.58316pt]{0.51663pt}{3.0998pt}}\rule[-2.06653pt]{0.51663pt}{2.58316pt}\rule{7.50002pt}{0.51663pt}\rule[-2.06653pt]{0.51663pt}{2.58316pt}$}}}{{\mbox{$\displaystyle\bm{\mathsf{A}}$}}}\hskip
1.99997pt$}\in\mathcal{T}(r,\mathsf{V})$, the representation formula in (237)
reads as

$\mbox{$\hskip
1.99997pt\stackrel{{\scriptstyle\mbox{$\vphantom{\rule[-2.58316pt]{0.51663pt}{3.0998pt}}\rule[-2.06653pt]{0.51663pt}{2.58316pt}\rule{7.50002pt}{0.51663pt}\rule[-2.06653pt]{0.51663pt}{2.58316pt}$}}}{{\mbox{$\displaystyle\bm{\mathsf{A}}$}}}\hskip
1.99997pt$}=A_{r}\mbox{$\hskip
1.99997pt\stackrel{{\scriptstyle\mbox{$\vphantom{\rule[-2.58316pt]{0.51663pt}{3.0998pt}}\rule[-2.06653pt]{0.51663pt}{2.58316pt}\rule{39.41751pt}{0.51663pt}\rule[-2.06653pt]{0.51663pt}{2.58316pt}$}}}{{\mbox{$\displaystyle\bm{a}_{1}\otimes\bm{a}_{2}\cdots\bm{a}_{r}$}}}\hskip
1.99997pt$},$ (238)

which easily identifies both the number $N(r)$ of independent parameters
needed to represent $\hskip
1.99997pt\stackrel{{\scriptstyle\mbox{$\vphantom{\rule[-2.58316pt]{0.51663pt}{3.0998pt}}\rule[-2.06653pt]{0.51663pt}{2.58316pt}\rule{7.50002pt}{0.51663pt}\rule[-2.06653pt]{0.51663pt}{2.58316pt}$}}}{{\mbox{$\displaystyle\bm{\mathsf{A}}$}}}\hskip
1.99997pt$ and all the invariants allowed in an isotropic scalar-valued
function of $\hskip
1.99997pt\stackrel{{\scriptstyle\mbox{$\vphantom{\rule[-2.58316pt]{0.51663pt}{3.0998pt}}\rule[-2.06653pt]{0.51663pt}{2.58316pt}\rule{7.50002pt}{0.51663pt}\rule[-2.06653pt]{0.51663pt}{2.58316pt}$}}}{{\mbox{$\displaystyle\bm{\mathsf{A}}$}}}\hskip
1.99997pt$.

Customarily, for $r\geqq 2$ and $\dim\mathsf{V}>2$, $N(r)$ is given by a
combinatoric argument as the difference between two binomial coefficients
(see, for example, [104, p. 56]),

$N(r)={r+2\choose r}-{r\choose r-2},$ (239)

the first representing the number of symmetric arrangements (with repetitions)
of $r$ symbols out of a pool of $3$, and the second the number of symmetric
arrangements (with repetitions) of $r-2$ symbols out of the same pool (these
latter corresponding to the number of traces). A simple calculation shows that
$N(r)=2r+1$. The same conclusion is reached far more easily from (238) by
remarking that $2r$ parameters are needed to represent the vectors
$\bm{a}_{1},\bm{a}_{2},\dots\bm{a}_{r}$ on $\mathbb{S}^{2}$ and one more for
$A_{r}$.

######  Remark 28.

The case $\dim\mathsf{V}=2$ is special. $N(r)$ is no longer given by (239),
but $N(r)=2$ for all $r$. Moreover, (238) is replaced by

$\mbox{$\hskip
1.99997pt\stackrel{{\scriptstyle\mbox{$\vphantom{\rule[-2.58316pt]{0.51663pt}{3.0998pt}}\rule[-2.06653pt]{0.51663pt}{2.58316pt}\rule{7.50002pt}{0.51663pt}\rule[-2.06653pt]{0.51663pt}{2.58316pt}$}}}{{\mbox{$\displaystyle\bm{\mathsf{A}}$}}}\hskip
1.99997pt$}=A_{r}\mbox{$\hskip
1.99997pt\stackrel{{\scriptstyle\mbox{$\vphantom{\rule[-2.58316pt]{0.51663pt}{3.0998pt}}\rule[-2.06653pt]{0.51663pt}{2.58316pt}\rule{26.07983pt}{0.51663pt}\rule[-2.06653pt]{0.51663pt}{2.58316pt}$}}}{{\mbox{$\displaystyle\underbrace{\bm{e}\otimes\cdots\otimes\bm{e}}_{r\
\mathrm{times}}$}}}\hskip 1.99997pt$},$ (240)

where all vectors $\bm{a}_{i}$ are the same vector $\bm{e}$ on the unit circle
$\mathbb{S}^{1}$ (see [117] for an explicit construction of $\bm{e}$ when
$r=3$).

Similarly, since $\hskip
1.99997pt\stackrel{{\scriptstyle\mbox{$\vphantom{\rule[-2.58316pt]{0.51663pt}{3.0998pt}}\rule[-2.06653pt]{0.51663pt}{2.58316pt}\rule{7.50002pt}{0.51663pt}\rule[-2.06653pt]{0.51663pt}{2.58316pt}$}}}{{\mbox{$\displaystyle\bm{\mathsf{A}}$}}}\hskip
1.99997pt$ is fully determined by $A_{r}$ and the multipoles
$\bm{a}_{1},\bm{a}_{2},\dots,\bm{a}_{r}$, the classical theorem of Cauchy [20]
(see also [112, p. 29]) for the representation of isotropic scalar-valued
functions depending on a finite number of vectors requires that the complete
list of invariants consists of $A_{r}$ and the following $r(r-1)/2$ scalars

$\alpha_{ij}:=\bm{a}_{i}\cdot\bm{a}_{j}\quad 1\leqq i<j\leqq r.$ (241)

Then, in the special case where $r=3$, the total number of scalar invariants
of $\hskip
1.99997pt\stackrel{{\scriptstyle\mbox{$\vphantom{\rule[-2.58316pt]{0.51663pt}{3.0998pt}}\rule[-2.06653pt]{0.51663pt}{2.58316pt}\rule{7.50002pt}{0.51663pt}\rule[-2.06653pt]{0.51663pt}{2.58316pt}$}}}{{\mbox{$\displaystyle\mathbf{A}$}}}\hskip
1.99997pt$ is immediately seen to be $4$, in agreement with [103] (see remark
1). Although the multipole representation of $\hskip
1.99997pt\stackrel{{\scriptstyle\mbox{$\vphantom{\rule[-2.58316pt]{0.51663pt}{3.0998pt}}\rule[-2.06653pt]{0.51663pt}{2.58316pt}\rule{7.50002pt}{0.51663pt}\rule[-2.06653pt]{0.51663pt}{2.58316pt}$}}}{{\mbox{$\displaystyle\bm{\mathsf{A}}$}}}\hskip
1.99997pt$ in (238) can more easily determine the _number_ of invariants,
their explicit identification may be more difficult.

######  Remark 29.

The number of isotropic invariants for symmetric traceless tensors
$\bm{\mathsf{D}}^{(3)}$ and $\bm{\mathsf{D}}^{(4)}$, of rank $3$ and $4$ in
three space dimensions, were derived in [103] and [13] from the determination
of the appropriate integrity bases, and found to be $4$ and $9$, respectively.
As shown in [131], the direct derivation of this number from the harmonic
decomposition in (238) agrees with that in [103] for $\bm{\mathsf{D}}^{(3)}$,
but it does _not_ with that in [103, 13] for $\bm{\mathsf{D}}^{(4)}$, as it
would predict $7$ invariants for the latter instead of $9$. This suggests that
the invariants in the integrity bases for $\bm{\mathsf{D}}^{(4)}$ in [103, 13]
are not independent. A table of other inconsistencies similar to this can be
found in [131]. A large number of studies are devoted to this issue (which is
not central to our review). Some, such as [115, 2], are especially relevant to
the mechanics of composite materials.

The octupolar potential $\Phi$ has played a special role in our review. We now
wish to show how $\Phi$ would be expressed for the harmonic representation in
(238) for an octupolar tensor $\hskip
1.99997pt\stackrel{{\scriptstyle\mbox{$\vphantom{\rule[-2.58316pt]{0.51663pt}{3.0998pt}}\rule[-2.06653pt]{0.51663pt}{2.58316pt}\rule{7.50002pt}{0.51663pt}\rule[-2.06653pt]{0.51663pt}{2.58316pt}$}}}{{\mbox{$\displaystyle\mathbf{A}$}}}\hskip
1.99997pt$ in three space dimensions. A direct computation based on (49) shows
that

$\qquad\eqalign{\Phi(\bm{x})&=A_{3}\Big{\\{}(\bm{a}_{1}\cdot\bm{x})(\bm{a}_{2}\cdot\bm{x})(\bm{a}_{3}\cdot\bm{x})\\\
&-\frac{1}{5}(\bm{x}\cdot\bm{x})\big{[}(\bm{a}_{1}\cdot\bm{x})(\bm{a}_{2}\cdot\bm{a}_{3})+(\bm{a}_{2}\cdot\bm{x})(\bm{a}_{3}\cdot\bm{a}_{1})+(\bm{a}_{3}\cdot\bm{x})(\bm{a}_{1}\cdot\bm{a}_{2})\big{]}\Big{\\}}.}$
(242)

It is instructive to see what becomes of $\Phi$ in (242) for special choices
of the unit vectors $\bm{a}_{1},\bm{a}_{2},\bm{a}_{3}$ and how these special
forms of $\Phi$ relate to those described above in our analysis.

First, given a Cartesian frame $(\bm{e}_{1},\bm{e}_{2},\bm{e}_{3})$, we
consider the case where $\bm{a}_{1}=\bm{a}_{2}=\bm{a}_{3}=\bm{e}_{3}$. It
follows from (242) that $\Phi$ then reduces to

$\Phi(\bm{x})=x_{3}\left(x_{3}^{2}-\frac{3}{2}x_{1}^{2}-\frac{3}{2}x_{2}^{2}\right),$
(243)

where we have set $A_{3}=5/2$ so that $\Phi(\bm{e}_{3})=1$. With this
normalization, the polar plot of $\Phi$ is just the same as the one in figure
2(a).

If we now take $\bm{a}_{i}=\bm{e}_{i}$, for $i=1,2,3$, (242) delivers

$\Phi(\bm{x})=3\sqrt{3}x_{1}x_{2}x_{3},$ (244)

where we have chosen $A_{3}=3\sqrt{3}$ so that the maximum value of $\Phi$ on
$\mathbb{S}^{2}$ be $\Phi=1$. Figure 27

Figure 27: Polar plot of the octupolar potential in (244). Its maxima and
minima fall on the tetrahedral vectors $\bm{n}_{\alpha}$ defined in (61) and
shown in figure 1.

illustrates the polar plot of the function in (244) in the conventional
representation adopted here: it only differs by a rigid rotation from the
polar plot shown in figure 5(a) corresponding to the tetrahedral symmetry
$T_{d}$ studied in section 3.4.

The minima and maxima of $\Phi$ in (244) are attained at the unit vectors
$\bm{n}_{\alpha}$ defined in (61) and illustrated in figure 1. This can also
be seen by considering the _tetrahedral_ tensor

$\mathbf{T}:=T\sum_{\alpha=1}^{4}\bm{n}_{\alpha}\otimes\bm{n}_{\alpha}\otimes\bm{n}_{\alpha},$
(245)

with $T$ a normalizing scalar. Since
$\sum_{\alpha=1}^{4}\bm{n}_{\alpha}=\bm{0}$, $\mathbf{T}$ is a symmetric
traceless octupolar tensor. The octupolar potential $\Phi_{\mathrm{T}}$
associated with it is given by

$\displaystyle\qquad\Phi_{\mathrm{T}}(\bm{x})$
$\displaystyle:=\mathbf{T}\cdot(\bm{x}\otimes\bm{x}\otimes\bm{x})=T\big{[}(\bm{n}_{1}\cdot\bm{x})^{3}+(\bm{n}_{2}\cdot\bm{x})^{3}+(\bm{n}_{3}\cdot\bm{x})^{3}+(\bm{n}_{4}\cdot\bm{x})^{3}\big{]}$
(246) $\displaystyle=-\frac{8T}{\sqrt{3}}x_{1}x_{2}x_{3},$

which reduces to (244) for $T=-9/8$. Comparing (244) and (210) would also be
instructive.

###  6.2 Curie potential

We have often said that the octupolar potential $\Phi$, which has been our
major tool in this review, identifies completely an octupolar tensor only if
this is fully symmetric. Thus, for example, the octupolar potential associated
with a piezoelectric tensor $\mathbf{A}$, as defined in section 2.4.3, would
fail to capture all its details. In particular, the definition of generalized
eigenvalues and eigenvectors of $\mathbf{A}$ given in section 2.3 would be
missed. A strategy has recently been developed in [24, 63, 64, 122] to
overcome this difficulty and to attempt at providing a similar treatment for
both piezoelectric and fully symmetric octupolar tensors. Here, we briefly
present this strategy, following mainly [24].

We start by defining the _Curie_ potential $\Phi_{\mathrm{C}}$ of a
piezoelectric tensor $\mathbf{A}$ in three space dimensions,161616The theory
summarized in [24] applies to a general piezoelectric tensor
$\bm{\mathsf{A}}\in\mathcal{T}(3,\mathsf{V})$ with $\dim\mathsf{V}=n$. Here,
we present the simplified version for $n=3$, as it is more germane to the rest
of our analysis.

$\Phi_{\mathrm{C}}(\bm{x},\bm{y}):=\mathbf{A}\cdot\bm{x}\otimes\bm{y}\otimes\bm{y},$
(247)

which is a mapping
$\Phi_{\mathrm{C}}:\mathbb{S}^{2}\times\mathbb{S}^{2}\to\mathbb{R}$. In
components relative to a Cartesian frame $(\bm{e}_{1},\bm{e}_{2},\bm{e}_{3})$,
equation (247) reads as

$\Phi_{\mathrm{C}}(\bm{x},\bm{y})=A_{ijk}x_{i}y_{j}y_{k}.$ (248)

Reasoning as in section 2.3 (see also equations (102) and (103)), the critical
points of $\Phi_{\mathrm{C}}$ on $\mathbb{S}^{2}\times\mathbb{S}^{2}$ can be
viewed as critical points of the unconstrained potential

$\displaystyle\Phi_{\lambda,\mu}(\bm{x},\bm{y}):=\Phi_{\mathrm{C}}(\bm{x},\bm{y})-\frac{1}{2}\lambda\left(x_{1}^{2}+x_{2}^{2}+x_{3}^{2}\right)-\mu\left(y_{1}^{2}+y_{2}^{2}+y_{3}^{2}\right),$
(249)

where $\lambda$ and $\mu$ are independent Lagrange multipliers. Such
unconstrained critical points are solutions of the following system of
equations,

$\cases{A_{ijk}y_{j}y_{k}=\lambda x_{i},\\\ A_{ijk}x_{i}y_{j}=\mu y_{k}.}$
(250)

Multiplying by $x_{i}$ the first and by $y_{k}$ the second (summing over
repeated indices), and enforcing the constraints that require both $\bm{x}$
and $\bm{y}$ on $\mathbb{S}^{2}$, we easily conclude that $\lambda=\mu$ and
their common value is precisely the value of $\Phi_{\mathrm{C}}$ at the
corresponding critical point. A pair $(\bm{x},\bm{y})$ that solves (250) is
said to consist of a _left_ and a _right C-eigenvector_ of $\mathbf{A}$,
respectively, and $\lambda=\mu$ is the corresponding _C-eigenvalue_ (here the
prefix _C_ stands for _Curie_) [24].

A number of facts have been established for the C-eigenvalues of a
piezoelectric octupolar tensor $\mathbf{A}$ (see Theorems 2.3 and 2.5 of
[24]):

  1. (I)

C-eigenvalues and associated left and right C-eigenvectors of $\mathbf{A}$ do
exist.

  2. (II)

If $\lambda$ is a C-eigenvalue of $\mathbf{A}$ and $(\bm{x},\bm{y})$ are the
corresponding left and right C-eigenvectors, then

$\mathbf{A}\cdot\bm{x}\otimes\bm{y}\otimes\bm{y}=\lambda.$ (251)

Moreover, the triples $(\lambda,\bm{x},-\bm{y})$, $(-\lambda,-\bm{x},\bm{y})$,
and $(-\lambda,-\bm{x},-\bm{y})$ also designate C-eigenvalues and
corresponding C-eigenvectors of $\mathbf{A}$.

  3. (III)

Let $\lambda_{1}$ denote the largest C-eigenvalue of $\mathbf{A}$ and let
$(\bm{x}_{1},\bm{y}_{1})$ denote the corresponding left and right
C-eigenvectors. Then

$\lambda_{1}=\max\\{\mathbf{A}\cdot\bm{x}\otimes\bm{y}\otimes\bm{y}:\bm{x},\bm{y}\in\mathbb{S}^{2}\\}.$
(252)

Moreover, $\lambda_{1}\bm{x}_{1}\otimes\bm{y}_{1}\otimes\bm{y}_{1}$ is the
rank-one tensor that best approximates $\mathbf{A}$, that is, it solves the
following optimization problem,

$\min\\{\|\mathbf{A}-\lambda\bm{x}\otimes\bm{y}\otimes\bm{y}\|^{2}:\lambda\in\mathbb{R},\
\bm{x},\bm{y}\in\mathbb{S}^{2}\\},$ (253)

where

$\|\mathbf{A}\|:=\sqrt{A_{ijk}A_{ijk}}$ (254)

designates the Frobenius norm.

  4. (IV)

If a piezoelectric tensor $\mathbf{A}$ has finitely many classes of
C-eigenvalues in the complex field $\mathbb{C}$, their number counted with
multiplicity is $13$.171717This applies to
$\mathbf{A}\in\mathcal{T}(3,\mathsf{V})$ with $\dim\mathsf{V}=3$. In general,
for $\dim\mathsf{V}=n$, this number is $(3^{n}-1)/2$, which is derived in [23]
by an extension of (54).

Property (III) establishes a connection between C-eigenvalues and the best
one-rank approximation of $\mathbf{A}$. We can think of applying recursively
this approximation algorithm, as suggested in [126] (where it was called
_incremental_ rank-one approximation), so that the second iterate would
deliver the best one-rank approximation
$\lambda_{2}\bm{x}_{2}\otimes\bm{y}_{2}\otimes\bm{y}_{2}$ to
$\mathbf{A}_{1}:=\mathbf{A}-\lambda_{1}\bm{x}_{1}\otimes\bm{y}_{1}\otimes\bm{y}_{2}$,
and so on; the existence of C-eigenvalues established in (I) guarantees that
this task can be accomplished at each step, up to the $p$-th iterate, when
$\mathbf{A}_{p}$ is itself rank-one.

According to the definition given in [126], a piezoelectric tensor
$\mathbf{A}$ is said to be _orthogonally decomposable_ if it can be written as
the following finite sum,

$\eqalign{\mathbf{A}=&\sum_{i=i}^{p}\lambda_{i}\bm{x}_{i}\otimes\bm{y}_{i}\otimes\bm{y}_{i},\
\lambda_{i}>0,\ \bm{x}_{i},\bm{y}_{i}\in\mathbb{S}^{2}\quad\\\ &\mathrm{with}\
\bm{x}_{i}\cdot\bm{x}_{j}=\bm{y}_{i}\cdot\bm{y}_{j}=0\quad\forall i\neq j.}$
(255)

It was proved in [126] that an orthogonally decomposable tensor $\mathbf{A}$
possesses a unique decomposition (255) and this is correctly identified by the
incremental rank-one approximation algorithm (a different proof of this result
can also be found in [62]).

######  Remark 30.

The _singular value decomposition_ of a second-rank tensor
$\bm{L}\in\mathcal{T}(2,\mathsf{V})$ with $\dim\mathsf{V}=n$ amounts to
represent it in the form

$\bm{L}=\bm{USV}^{\mathsf{T}},$ (256)

where

$\bm{S}=\sum_{i=1}^{n}\sigma_{i}\bm{e}_{i}\otimes\bm{e}_{i}\quad\mathrm{with}\quad\sigma_{i}\geqq
0\quad\mathrm{and}\quad\bm{e}_{i}\cdot\bm{e}_{j}=\delta_{ij}$ (257)

and $\bm{U}$, $\bm{V}$ are orthogonal tensors (such that
$\bm{UU}^{\mathsf{T}}=\bm{VV}^{\mathsf{T}}=\bm{I}$). This result has a long
history (neatly recounted in [106]) that started with the works of Beltrami
[10] and Jordan [57, 58]. What makes it relevant to our topic is that (256)
was also proved in [33] as resulting from a rank-one approximation of
$\bm{L}$, much in the same spirit as (255), which could thus be seen as a
possible extension of (256).

######  Remark 31.

The appropriate version of (255) valid for a generic orthogonally decomposable
octupolar tensor $\mathbf{A}$ is

$\eqalign{\mathbf{A}=&\sum_{i=i}^{p}\lambda_{i}\bm{x}_{i}\otimes\bm{y}_{i}\otimes\bm{z}_{i},\
\lambda_{i}>0,\ \bm{x}_{i},\bm{y}_{i},\bm{z}_{i}\in\mathbb{S}^{2}\\\
&\mathrm{with}\
\bm{x}_{i}\cdot\bm{x}_{j}=\bm{y}_{i}\cdot\bm{y}_{j}=\bm{z}_{i}\cdot\bm{z}_{j}=0\
\forall i\neq j.}$ (258)

The applicability of the incremental rank-one approximation algorithm to
establish (258) was also proved in [126]

######  Remark 32.

Introducing for a general octupolar tensor $\mathbf{A}$ the generalized
potential

$\Phi_{\mathrm{G}}(\bm{x},\bm{y},\bm{z}):=\mathbf{A}\cdot\bm{x}\otimes\bm{y}\otimes\bm{z},$
(259)

one could easily justify the rank-one approximation algorithm delivering (258)
as resulting from the search for the maximum of $\Phi_{\mathrm{G}}$ over
$\mathbb{S}^{2}\times\mathbb{S}^{2}\times\mathbb{S}^{2}$, which entrains a
further generalized notion of eigenvalues and associated eigenvectors
$(\lambda,\bm{x},\bm{y},\bm{z})$ of $\mathbf{A}$.

######  Remark 33.

Even when the orthogonal decompositions in (255) and (258) do not apply, the
one-rank approximation algorithm is still meaningful. In that case, the
orthogonality conditions in both (255) and (258) fail to hold and the
decompositions formally delivered by these equations no longer represent
$\mathbf{A}$; they feature the best approximations to $\mathbf{A}$ provided by
its generalized eigenvalues and eigenvectors.

##  7 Selected Applications

The applications of octupolar tensors in physics are countless. Apart from the
specific fields that in section 2.4 served as our motivation for this review,
other fields have witnessed new or renewed formulations of theories that use
octupolar (as well as higher-rank tensors). Here we give short accounts on
just exemplary few of these fields, pausing longer on liquid crystal science,
which is where our interest on the topic of this review originated.

###  7.1 Gravitation

In this context, octupolar tensors appear in the description of cubic-order
spin effects in the dynamics of gravitational waves [77]. Also, they feature
in computing invariants connected with tidal interactions that influence the
late dynamics of compact binary systems, which have the potential of
constituting the prime targets of a network of gravitational-wave detectors
[11].

###  7.2 Spin states

Majorana [74] introduced a geometrical picture to represent quantum states. In
this representation, a pure spin-$j$ state is mapped onto $2j$ points on the
unit sphere $\mathbb{S}^{2}$ (which is in this context is also called the
_Bloch sphere_). Recently, a generalization of this picture was proposed in
[47], which applies to both pure and mixed spin-$j$ states; this extended
representation employs a symmetric tensor of rank $2j$ in dimension $4$ (which
is thus an octupolar tensor for fermions with $j=3/2$). Along the same lines,
the reader will find it useful to consult the works [48, 15, 91].

###  7.3 Liquid crystals

In classical liquid crystal theory, the nematic director field $\bm{n}$
describes the average orientation of the molecules that constitute the medium;
the elastic distortions of $\bm{n}$ are locally measured by its gradient
$\nabla\bm{n}$, which may become singular where the director exhibits
_defects_ arising from a degradation of molecular order. The orientation of
$\bm{n}$ should be physically indistinguishable from the orientation of
$-\bm{n}$; this notion of invariance embodies the _nematic_ symmetry. In this
short account we follow [82], to which the reader is referred for any further
details.

The two main descriptors, $\bm{n}$ and $\nabla\bm{n}$, can be combined into
the third-rank octupolar tensor

$\mathbf{A}:=\mbox{$\hskip
1.99997pt\stackrel{{\scriptstyle\mbox{$\vphantom{\rule[-2.58316pt]{0.51663pt}{3.0998pt}}\rule[-2.06653pt]{0.51663pt}{2.58316pt}\rule{29.78249pt}{0.51663pt}\rule[-2.06653pt]{0.51663pt}{2.58316pt}$}}}{{\mbox{$\displaystyle\nabla\bm{n}\otimes\bm{n}$}}}\hskip
1.99997pt$},$ (260)

It is worth noticing that $\mathbf{A}$ defined in (260) is invariant under the
change of orientation of $\bm{n}$, and so it duly enjoys the nematic symmetry,
which makes it a good candidate for measuring intrinsically the local
distortions of a director field.

Selinger [101], extending earlier work [71], suggested a new interpretation of
the elastic modes for nematic liquid crystals described by the Oseen-Frank
elastic free energy, which penalizes in a quadratic fashion the distortions of
$\bm{n}$ away from any uniform state. The Oseen-Frank energy density
$W_{\mathrm{OF}}$ is defined as (see, e.g., [31, Chap. 3] and [116, Chap. 3])

$\eqalign{W_{\mathrm{OF}}&:=\frac{1}{2}K_{11}(\mathrm{div}\,\bm{n})^{2}+\frac{1}{2}K_{22}(\bm{n}\cdot\mathrm{curl}\,\bm{n})^{2}+\frac{1}{2}K_{33}|\bm{n}\times\mathrm{curl}\,\bm{n}|^{2}\\\
&\ +K_{24}[\tr(\nabla\bm{n})^{2}-(\mathrm{div}\,\bm{n})^{2}],}$ (261)

where $K_{11}$, $K_{22}$, $K_{33}$, and $K_{24}$ are the _splay_ , _twist_ ,
_bend_ , and _saddle-splay_ constants, respectively, each associated with a
corresponding elastic mode.181818The saddle-splay term is a null Lagrangian
[38] and an integration over the bulk reduces it to a surface energy. Here,
however, the surface-like nature of $K_{24}$ will not be exploited.

The decomposition of $W_{\mathrm{OF}}$ in independent elastic modes proposed
in [101] is achieved through a new decomposition of $\nabla\bm{n}$. If we
denote by $\mathbf{P}(\bm{n})$ and $\mathbf{W}(\bm{n})$ the projection onto
the plane orthogonal to $\bm{n}$ and the skew-symmetric tensor with axial
vector $\bm{n}$, respectively, then

$\nabla\bm{n}=-\bm{b}\otimes\bm{n}+\frac{1}{2}T\mathbf{W}(\bm{n})+\frac{1}{2}S\mathbf{P}(\bm{n})+\mathbf{D},$
(262)

where $\bm{b}:=-(\nabla\bm{n})\bm{n}=\bm{n}\times\mathrm{curl}\,\bm{n}$ is the
_bend_ vector, $T:=\bm{n}\cdot\mathrm{curl}\,\bm{n}$ is the _twist_ (a
pseudoscalar), $S:=\mathrm{div}\,\bm{n}$ is the _splay_ (a scalar), and
$\mathbf{D}$ is a symmetric tensor such that $\mathbf{D}\bm{n}=\bm{0}$ and
$\tr\mathbf{D}=0$. The properties of $\mathbf{D}$ guarantee that when
$\mathbf{D}\neq\bm{0}$ it can be represented as

$\mathbf{D}=q\left(\bm{n}_{1}\otimes\bm{n}_{1}-\bm{n}_{2}\otimes\bm{n}_{2}\right),$
(263)

where $q$ is the _positive_ eigenvalue of $\mathbf{D}$. We shall call $q$ the
_octupolar splay_ for a reason that shall soon be clear. The choice of sign
for $q$ identifies (to within the orientation) the eigenvectors $\bm{n}_{1}$
and $\bm{n}_{2}$ of $\mathbf{D}$ orthogonal to $\bm{n}$. Since
$\tr\mathbf{D}^{2}=2q^{2}$, we easily obtain from (262) that

$2q^{2}=\tr(\nabla\bm{n})^{2}+\frac{1}{2}T^{2}-\frac{1}{2}S^{2}.$ (264)

$W_{\mathrm{OF}}$ can then be given the form

$W_{\mathrm{OF}}=\frac{1}{2}(K_{11}-K_{24})S^{2}+\frac{1}{2}(K_{22}-K_{24})T^{2}+\frac{1}{2}K_{33}b^{2}+K_{24}(2q^{2}),$
(265)

where all quadratic contributions are independent from one another.

The first advantage of such an expression is that it explicitly shows when the
free energy is positive semi-definite; this is the case when the following
inequalities, due to Ericksen [39], are satisfied,

$K_{11}\geqq K_{24}\geqq 0,\quad K_{22}\geqq K_{24}\geqq 0,\quad K_{33}\geqq
0.$ (266)

Whenever $q>0$, the frame $(\bm{n}_{1},\bm{n}_{2},\bm{n})$ is identified to
within a change of sign in either $\bm{n}_{1}$ or $\bm{n}_{2}$; requiring that
$\bm{n}=\bm{n}_{1}\times\bm{n}_{2}$, we reduce this ambiguity to a
simultaneous change in the orientation of $\bm{n}_{1}$ and $\bm{n}_{2}$. In
this frame,

$\mathbf{P}(\bm{n})=\mathbf{I}-\bm{n}\otimes\bm{n}\quad\mathrm{and}\quad\mathbf{W}(\bm{n})=\bm{n}_{2}\otimes\bm{n}_{1}-\bm{n}_{1}\otimes\bm{n}_{2}.$
(267)

Since $\bm{b}\cdot\bm{n}\equiv 0$, we can represent $\bm{b}$ as
$\bm{b}=b_{1}\bm{n}_{1}+b_{2}\bm{n}_{2}$. The frame
$(\bm{n}_{1},\bm{n}_{2},\bm{n})$ is called the _distortion frame_ and
$(S,T,b_{1},b_{2},q)$ the _distortion characteristics_ of the director field
$\bm{n}$ [118]. In terms of these, (262) can also be written as

$\qquad\eqalign{\nabla\bm{n}&=\left(\frac{S}{2}+q\right)\bm{n}_{1}\otimes\bm{n}_{1}+\left(\frac{S}{2}-q\right)\bm{n}_{2}\otimes\bm{n}_{2}-b_{1}\bm{n}_{1}\otimes\bm{n}-b_{2}\bm{n}_{2}\otimes\bm{n}\\\
&+\frac{1}{2}T\left(\bm{n}_{2}\otimes\bm{n}_{1}-\bm{n}_{1}\otimes\bm{n}_{2}\right).}$
(268)

Both (262) and (268) show an intrinsic decomposition of $\nabla\bm{n}$ into
four genuine bulk contributions, namely, bend, splay, twist, and octupolar
splay.

The octupolar tensor $\mathbf{A}$ defined in (260) revealed itself as a
convenient tool to illustrate director distortions [82]. Having, however,
symmetrized $\mathbf{A}$, we have implicitly renounced to represent $T$, so no
sign of twist will be revealed by $\mathbf{A}$. This is the only piece of lost
information.

######  Remark 34.

$T$ is a measure of chirality, and so it cannot be associated with a symmetric
tensor. By forming the completely skew-symmetric part of
$\nabla\bm{n}\otimes\bm{n}$, one would obtain the tensor
$-\frac{1}{6}T\bm{\epsilon}$, where $\bm{\epsilon}$ is Ricci’s alternator, the
most general skew-symmetric, third-rank tensor in three dimensions.

Letting $\bm{x}=x_{1}\bm{n}_{1}+x_{2}\bm{n}_{2}+x_{3}\bm{n}$ be a point on the
unit sphere $\mathbb{S}^{2}$ referred to the distortion frame
$(\bm{n}_{1},\bm{n}_{2},\bm{n})$, with the aid of (268), the octupolar
potential $\Phi$ defined by (82) can be written for $\mathbf{A}$ in (260) as
follows

$\eqalign{\Phi(\bm{x})&=\left(\frac{S}{2}+q\right)x_{1}^{2}x_{3}+\left(\frac{S}{2}-q\right)x_{2}^{2}x_{3}-b_{1}x_{1}x_{3}^{2}-b_{2}x_{2}x_{3}^{2}\\\
&+\frac{1}{5}\big{(}x_{1}^{2}+x_{2}^{2}+x_{3}^{2}\big{)}\big{(}b_{1}x_{1}+b_{2}x_{2}-Sx_{3}\big{)}.}$
(269)

As expected, $\Phi$ does not depend on the twist $T$, but it does depend on
the octupolar splay $q$.

A thorough analysis of $\Phi$ in (269) is performed in [82]. Here, we only
describe the very special cases where one and only one elastic mode is
exhibited.

#### Splay.

When splay is the only active mode, the choice of $\bm{n}_{1}$ and
$\bm{n}_{2}$ in the plane orthogonal to $\bm{n}$ is arbitrary. This fact
reverberates in the symmetries of the octupolar potential and also in its
critical points. In this case,

$\Phi(\bm{x})=\frac{1}{10}S\big{(}3x_{1}^{2}x_{3}+3x_{2}^{2}x_{3}-2x_{3}^{3}\big{)}.$
(270)

Graphically, $\Phi(\bm{x})$ is depicted in figure 28(a), which is nothing but
figure 2(a) turned upside down.

(a) Pure splay (b) Pure octupolar splay (c) Pure bend

Figure 28: Polar plots of the octupolar potential $\Phi$ in (269) for pure
elastic modes. Dashed lines are associated with maxima (and conjugated
minima). Reprinted with permission from [82].

#### Octupolar splay.

When both $S=0$ and $b=0$, but $q>0$, the potential $\Phi$ reduces to

$\Phi(\bm{x})=q(x_{1}^{2}-x_{2}^{2})x_{3}.$ (271)

Figure 28(b) shows that $\Phi(\bm{x})$ has four identical lobes, spatially
distributed at the vertices of a regular tetrahedron; this is just the same
plot as in figures 5(a) and 27, but differently oriented in the reference
frame. Accordingly, its maxima are the four points

$\bm{x}_{1,2}=\frac{1}{\sqrt{3}}(\pm\sqrt{2}\bm{n}_{1}+\bm{n})\quad\mathrm{and}\quad\bm{x}_{3,4}=\frac{1}{\sqrt{3}}(\pm\sqrt{2}\bm{n}_{2}-\bm{n}),$
(272)

each with value $2q/(3\sqrt{3})$.

#### Bend.

For pure bend, we can choose $\bm{n}_{1}$ and $\bm{n}_{2}$ such that
$\bm{b}=b\bm{n}_{1}$ with $b>0$. Then the potential,

$\Phi(\bm{x})=\frac{1}{5}bx_{1}\left(x_{1}^{2}+x_{2}^{2}-4x_{3}^{2}\right)=\frac{b}{5}x_{1}\left(1-5x_{3}^{2}\right),$
(273)

has three lobes: two larger, with equal height $16b/(15\sqrt{15})$ at

$\bm{x}_{1,2}=\frac{1}{\sqrt{15}}\left(-2\bm{n}_{1}\pm\sqrt{11}\bm{n}\right),$
(274)

and one smaller at $\bm{x}_{3}=\bm{n}_{1}$ with height $b/5$. As shown in
figure 28(c), the polar plot of $\Phi$ is invariant under both a rotation by
angle $\pi$ around $\bm{n}_{1}$ and the mirror symmetry with respect to the
plane containing $(\bm{n}_{1},\bm{n})$.

######  Remark 35.

In a phenomenological theory for modulated nematic liquid crystal phase
recently proposed in [98], octupolar order plays a central role, as molecules
are envisioned as stretched tetrahedra. Motivated in part by the properties of
the distortion tensor $\mathbf{D}$ in (263), the authors of this study
describe octupolar order through a third-rank tensor, which in our formalism
can be written as

$\mathbf{A}=\bm{\Omega}\otimes\bm{n},$ (275)

where $\bm{n}$ is the nematic director and $\bm{\Omega}$ is a second-rank
symmetric traceless tensor that annihilates $\bm{n}$. The tensor $\mathbf{A}$
in (275) falls in yet another category of third-rank tensors, which we have
not explicitly considered, but is amenable to the method outlined here. In
three space dimensions, this tensor is represented by $4$ scalar parameters;
it can be associated with the following octupolar potential on
$\mathbb{S}^{2}\times\mathbb{S}^{2}$,

$\Phi_{\mathrm{O}}(\bm{x},\bm{y}):=\mathbf{A}\cdot\bm{x}\otimes\bm{x}\otimes\bm{y}=(\bm{x}\cdot\bm{\Omega}\bm{x})(\bm{n}\cdot\bm{y}).$
(276)

##  8 Conclusion

Strictly speaking, an _octupolar tensor_ $\mathbf{A}$ is a third-rank
symmetric traceless tensor, which is also called a _harmonic_ tensor in some
literature. There is an impressive body of works devoted to this special class
of tensors and their application to diverse fields of physics. Here, we
endeavoured to review some of these works in an attempt to broaden the scope
where this specific mathematical tool can be placed.

Not only have we considered fully symmetric tensors, but also partly symmetric
ones and fully general tensors. Of course, the more general was the setting,
the less simple were the results.

In the diverse territories we have traversed we found guidance in the unifying
concept of _octupolar potential_ $\Phi$, which, being a scalar-valued function
representable on the unit sphere, added geometrical charm to a somewhat algid
algebra.

Seeing diverse approaches displayed before us, a number of questions come
naturally to mind, none necessarily with an easy answer. Many—we are sure—have
already been heeded by the reader. Here, we mention just two of these, which
have especially attracted our attention.

First, one wonders whether there is a systematic way to relate the generalized
eigenvectors of $\mathbf{A}$ to its multipoles. Second, one would like to
explore further the geometric properties enjoyed by the octupolar potential
$\Phi$ defined for a non-symmetric $\mathbf{A}$.

We hope that these and other issues may be addressed in the future as a result
of our attempt to put octupolar tensors within a unifying setting. We trust
that practitioners from the diverse fields touched upon in this review may
take even a modest advantage from the perspectives we have offered. Should
this be the case, our effort would not have been completely in vain.

We are grateful to Rebecca Gillan from IOP for having invited this review and
for her patience in tolerating the long delays that this project has suffered
from various interferences; her kind perseverance has been one of the major
drives for the completion of this work. Both authors are members of the
Italian _Gruppo Nazionale per la Fisica Matematica_ (GNFM), an articulation of
the Italian _Istituto Nazionale di Alta Matematica_ (INdAM). G.G. thanks the
_Santa Marinella Research Institute_ (SMRI), where his part of the present
work was carried out.

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