
\bigbreak
{\nobigbf 3. Gosset polytopes}

\nobreak\bigskip\nobreak

In this section, we compute the spectra of 
the {\sl Gosset polytopes} $G_n$.
The solid $G_3$ is the triangular prism in dimension 3
and the vertex figure of $G_n$ is $G_{n-1}$.
Coordinates for the vertices of the Gosset polytopes 
are given in [G], [C] and [C3].

The polytope $G_n$ has for faces simplices $\alpha_{n-1}$
and cross polytopes $\beta_{n-1}$.
Recall that a cross polytope is a generalized octahedron,
or $\{3,\ldots,3,4\}$ in Schl\"afli's notation;
the vertices of $\beta_n$ consist of a north and south pole
together with the vertices of an equatorial $\beta_{n-1}$.
We make extensive use of Table 3.1 ([C], section 11.8).

\midinsert\smallskip

\centerline{\vbox{\offinterlineskip\halign{%
\strut\quad\hfil#\hfil\quad&\vrule#&&\strut\quad\hfil#\hfil\quad\cr
&depth 8pt&$G_4$&$G_5$&$G_6$&$G_7$&$G_8$\cr
%\omit&height2pt&\omit&\omit&\omit&\omit&\omit&\cr
\noalign{\hrule}
%\omit&height2pt&\omit&\omit&\omit&\omit&\omit&\cr
Number of vertices&height 12pt&10&16&27&56&240\cr
Number of $\beta_{n-1}$ faces&&5&10&27&126&2160\cr}}}

\smallskip

\centerline{\bf Table 3.1}

\smallskip\endinsert

A more combinatorial description of the graphs
$G_i$, $i = 4,\ldots,8$, is given in [BCN]
where the notation $E_i(1)$
(reminiscent of the Lie algebras $E_n$) is employed.

Let $\Gamma(G_n)$ be the group of isometries of $G_n$
and, for an arbitrary choice of a vertex $p_1$ of $G_n$,
call $\Gamma_1(G_n)$ the subgroup of $\Gamma(G_n)$ fixing $p_1$.
As explained in [C3], $\Gamma_1(G_n) = \Gamma(G_{n-1})$.
Also, $\Gamma(G_n)$ acts transitively both on
the set of simplicial faces and
on the set of cross polytope faces.
The orbits of the action of $\Gamma_1(G_n)$
on the set of vertices of $G_n$ are called
{\sl suborbits} in the terminology of association schemes
and {\sl levels} in [ST].
The vertex $p_1$ forms a suborbit by itself;
the next suborbit is a $G_{n-1}$.
As we shall see, the number of suborbits of $G_n$
is 3, 3, 3, 4 and 5 for $n = 4$, 5, 6, 7 and 8.
Thus, the method employed in [ST] for regular polytopes
is well suited for the Gosset family.
Actually, this method is a special case of the technique
of {\sl regular partitions} ([BCN]) for the computation
of the spectrum of an association scheme.

We briefly describe this special method.
Let $v \ne 0$ satisfy $X v = \lambda v$
for the adjacency matrix $X$ of a semi-regular polytope.
Without loss, the value of $v$ at $p_1$ is nonzero.
Let $S$ be the space of vectors which are constant on suborbits:
$S$ is invariant under $X$.
The (nonzero) average $s \in S$ of $v$ under the action of $\Gamma_1$
is another eigenvector of $X$ associated
to the same eigenvalue $\lambda$.
Thus, up to multiplicities, the spectrum of $X$ equals
the spectrum of its restriction $B$ to $S$.
For the basis of $S$ consisting 
of vectors $s_k$ taking the value 1 at the $k$-th suborbit
and 0 elsewhere, the entry $b_{ij}$ of $B$
is the number of neighbours in suborbit $j$ of a vertex in suborbit $i$.
We now obtain the matrices $B_n$ for $G_n$, $n \ge 4$.

The 16 vertices of $G_5$ are $(\pm 1,\ldots,\pm 1)$
with an even number of minus signs ([C]).
Taking $p_1 = (1,\ldots,1)$, different sums of coordinates
clearly imply different suborbits.
The 3 possible values of this sum, 5, 1 and $-3$,
correspond to sets of 1, 10 and 5 vertices
which might, in principle, be the union of more than one suborbit.
The intermediate set is a $G_4$,
on which $\Gamma_1(G_5) = \Gamma(G_4)$ is transitive:
this set is therefore a suborbit.
Each of the five $\beta_3$ faces of the second suborbit
is an equator of a $\beta_4$ face of $G_5$
with north pole $p_1$ and south pole in the third set.
Since $\Gamma(G_4)$ acts transitively
on the cross polytope faces of $G_4$, this bottom set is a suborbit.
The $3 \times 3$ matrix $B_5$ is
$$B_5 = \pmatrix{0 & 10 & 0 \cr 1 & 6 & 3 \cr 0 & 6 & 4 \cr}$$
with eigenvalues 10, 2 and $-2$.
Let their multiplicities in $X$ be 1, $a$ and $b$
(recall that the largest eigenvalue is always simple).
We must then have $1 + a + b = 16$ and
$1 \cdot 10 + a \cdot 2 + b \cdot (-2) = \tr(X) = 0$;
$a$ and $b$ are then 5 and 10.

The central suborbit of $G_5$ gives coordinates (in $\RR^5$)
for $G_4$ and, as above, it is easy to obtain
the decomposition of $G_4$ into suborbits of 1, 6 and 3 elements,
yielding
$$B_4 = \pmatrix{0 & 6 & 0 \cr 1 & 3 & 2 \cr 0 & 4 & 2 \cr},$$
from which the spectrum of $G_4$ follows.

From coordinates for $G_5$,
one obtains coordinates for a $G_6$ inscribed in $S^6$
for which suborbits are indicated by the first entry.
Start with $p_1 = (1,0,\ldots,0)$ and add the next suborbit of vertices,
of the form $(a, b\,G_5)$,
where $a = 1/4$ and $b = (\sqrt{3}/4)$;
the coefficients are taken so that the second suborbit
lies in the unit sphere and all edges have equal length.
The vertices of a $\beta_5$ face of $G_6$ containing $p_1$
are, besides $p_1$, the vertices of a $\beta_4$ face
of the second suborbit (a $G_5$) and the reflection of $p_1$ on
the hyperplane containing this $\beta_4$.
We thus obtain 10 new vertices of $G_6$, with first coordinate $-1/2$,
associated to the 10 $\beta_4$ faces of the second suborbit;
$\Gamma_1(G_6)$ is transitive on the set of $\beta_4$ faces
of the second suborbit and these 10 points form a third suborbit of $G_6$.
Since $G_6$ has 27 vertices, these are the only suborbits;
adjacencies among suborbits are given by
$$B_6 = \pmatrix{0 & 16 & 0 \cr 1 & 10 & 5 \cr 0 & 8 & 8 \cr}$$
and we are done with $G_6$.

In a similar fashion, coordinates for $G_7$
in which the first entry indicates the suborbit are
$$(1,0,\ldots,0),\;(1/3,(2\sqrt{2}/3)\,G_6),\;%
(-1/3,-(2\sqrt{2}/3)\,G_6),\;(-1,0,\ldots,0).$$
The polytope $G_7$ then admits an involution taking $v$ to $-v$
splitting $V_{G_7}$ into two 28-dimensional subspaces
$V_+$ and $V_-$, invariant under $X$, and
$S$ into 2-dimensional subspaces $S_+$ and $S_-$, invariant under $B_7$.
The two restrictions of $B_7$ are
$$B_+ = \pmatrix{0 & 27 \cr 1 & 26 \cr},\quad
B_- = \pmatrix{0 & 27 \cr 1 & 6 \cr},$$
with spectra 27, $-1$ and 9, $-3$.
Considered as eigenvalues of $X$,
27 and $-1$ have multiplicities 1 and 27
since their eigenspaces are contained in $V_+$.
The two remaining multiplicities are now computed
taking into account that $\tr(X) = 0$.

Finally, analogous coordinates for $G_8$ are
$$(1,0,\ldots,0),\;(1/2,(\sqrt{3}/2)\,G_7),\;%
(0,\ast),\;(-1/2,(\sqrt{3}/2)\,G_7),\;(-1,0,\ldots,0),$$
where the vertices of the central suborbit
correspond to the 126 $\beta_6$ faces of the second suborbit (a $G_7$).
Again, $G_8$ admits an antipodal involution and $B_8$ splits as
$$B_+ = \pmatrix{0 & 56 & 0 \cr 1 & 28 & 27 \cr 0 & 24 & 32 \cr},\quad
B_- = \pmatrix{0 & 56 \cr 1 & 26 \cr}.$$
To compute the multiplicities, we only need the additional remark
that, since antipodal points are never neighbours,
the restrictions of $X$ to the 120-dimensional spaces $V_+$ and $V_-$
have trace zero.
