
\bigbreak
{\nobigbf 6. Three discrete subgroups of $S^3$}

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Among the six regular polytopes in four dimensions,
three of them, $\{3,3,4\}$, $\{3,4,3\}$ and $\{3,3,5\}$,
have a most remarkable property ([C], [C2]):
when suitably inscribed in the unit quaternionic sphere $S^3$,
their vertices form finite groups.
In this section, we briefly describe these three groups,
to be called $Q_8$, $Q_{24}$ and $Q_{120}$.

Following [C], we display appropriate choices
of unit quaternions for the vertices of these polytopes.
We identify $(a,b,c,d)$ to $a + bi + cj + dk$.
The elements of $Q_8$ are the vertices of $\{3,3,4\}$
with coordinates $\pm 1$, $\pm i$, $\pm j$ and $\pm k$.
In order to obtain the vertices of $\{3,4,3\}$,
add to the above list all 16 points of the form
$(\pm 1 \pm i \pm j \pm k)/2$, producing $Q_{24}$.
Finally, consider the 96 points obtained by even permutations of
the coordinates of $(\pm\tau,\pm 1,\pm\taub,0)/2$:
adding these to the 24 points already defined,
we obtain the vertices of $\{3,3,5\}$, i.e.,
the elements of $Q_{120}$.

The four groups $S^3$, $Q_8$, $Q_{24}$ and $Q_{120}$
have the same centre $\{\pm 1\}$.
As is well known ([MT]),
the quotient $S^3/\{\pm 1\}$ is isomorphic to $SO(3)$.
The quotient $Q_8/\{\pm 1\} = \ZZ/(2) \times \ZZ/(2)$,
as a group of isometries in $\RR^3$,
is generated by $180^\circ$ rotations around the three axis.
Similarly, $Q_{24}/\{\pm 1\} = A_4$ and
$Q_{120}/\{\pm 1\} = A_5$ are the groups of orientation preserving
isometries of a tetrahedron and an icosahedron, respectively.

Conjugacy classes in $S^3$ are determined by the first coordinate,
i.e., the real part;
in the groups $Q_n$, therefore,
points with different real parts are never conjugate.
The conjugacy classes of $Q_8$ are
$\{1\}$, $\{\pm i\}$, $\{\pm j\}$, $\{\pm k\}$, $\{-1\}$.
Representatives for the conjugacy classes of $Q_{24}$ are
1, $(1+i+j+k)/2$, $(1+i+j-k)/2$, $i$,
$(-1+i+j+k)/2$, $(-1+i+j-k)/2$ and $-1$;
these classes have 1, 8, 8, 6, 8, 8 and 1 elements, respectively.
For $Q_{120}$, the real part determines the conjugacy class:
there are thus 9 conjugacy classes denoted by
$1$, $b$, $c$, $d$, $e$, $-d$, $-c$, $-b$, $-1$ (decreasing real parts)
with 1, 12, 20, 12, 30, 12, 20, 12, 1 elements, respectively;
the order of an element in each conjugacy class is
1, 10, 6, 5, 4, 10, 3, 5, 2.
The conjugacy class $b$ consists of the 12 neighbours
of the vertex 1 and these 12 points form the vertices
of an icosahedron.

The irreducible representations of $S^3$ are well known ([Su]):
we call them $\sR_n$, where $\sR_n$ has dimension $n+1$.
The representation $\sR_0$ is trivial and
$\sR_1$ corresponds to the identification $S^3 = SU(2)$.
A basis for the space where $\sR_n$ acts is given by monomials
of degree $n$ in two variables $e_1$ and $e_2$,
identified with the basis of $\CC^2$;
for a homogeneous polynomial $\phi$ of degree $n$, define
$\sR_n(g)(\phi(e1,e2)) = \phi(\sR_1(g)(e_1),\sR_1(g)(e_2))$.
Restrictions of $\sR_n$ to $Q_n$ are still representations,
but are usually not irreducible.
Call an irreducible representation of $S^3$ or $Q_n$
{\sl even} (resp., {\sl odd}) if $-1$ is taken to $I$ (resp., $-I$);
$\sR_n$ is even if and only if $n$ is even.
The number of even (resp. odd) irreducible representations of
$Q_8$, $Q_{24}$ and $Q_{120}$ are 4, 4, 5 (resp. 1, 3, 4);
character tables for $Q_8$ and $Q_{24}$ are easy to obtain
and that of $Q_{120}$ is in [CCNPW].

We show how to obtain a matrix form for $\sM_i(G)$,
the $i$-th irreducible representation of $G$
(even representations come first;
within each parity, representations are ordered by dimension).
By restrictions, $\sR_1$ yield
$\sM_5(Q_8)$, $\sM_5(Q_{24})$ and $\sM_6(Q_{120})$.
Also, $\sR_2$ yields $\sM_4(Q_{24})$ and $\sM_2(Q_{120})$;
$\sR_3$, $\sR_4$ and $\sR_5$ yield
$\sM_8(Q_{120})$, $\sM_5(Q_{120})$ and $\sM_9(Q_{120})$.
The other representations can be obtained by algebraic conjugation
and tensor products:
$\sM_3(Q_{120})$ and $\sM_7(Q_{120})$ are the conjugates of
$\sM_2(Q_{120})$ and $\sM_6(Q_{120})$, respectively.
Also, $\sM_6(Q_{24}) = \sM_2(Q_{24}) \otimes \sM_5(Q_{24})$,
$\sM_7(Q_{24}) = \sM_3(Q_{24}) \otimes \sM_5(Q_{24})$ and
$\sM_4(Q_{120}) = \sM_6(Q_{120}) \otimes \sM_7(Q_{120})$.

In our examples, we frequently consider isometry groups in $\RR^4$.
As is well known ([MT]), $SO(4) = (S^3 \times S^3)/(-1,-1)$
by unit quaternion bilateral multiplication:
$(q,r) \cdot v = qvr^{-1}$.
Thus, the finite groups $(Q_n \times Q_m)/(-1,-1)$
act on $\RR^4$ by isometries.
The irreducible representations of such groups
are obtained by tensoring representations of each factor
having the same parity.

