
\bigbreak
{\nobigbf 4. Groups, Cayley graphs and representations}

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By using representations of groups,
Lov\'asz ([L]) described a method to compute
the spectrum of a Cayley graph, or, more generally,
of a graph with a transitive group of isomorphisms.
Babai ([B]) modified Lov\'asz's method
(and corrected a minor mistake in [L]) and
obtained families of Cayley graphs with the same spectrum.
We present yet another technique based on representations of groups
which is more convenient for our purposes.

For an undirected graph $\sG$, the {\sl doubling} $\tilde\sG$ of $\sG$
is the directed graph obtained by substituting
each edge of $\sG$ by two edges with opposite orientations.
A {\sl Cayley structure} for an undirected graph $\sG$
consists of a choice of a vertex (to be the identity)
and a colouring of the edges of the doubling $\tilde\sG$ of $\sG$
(to represent the set $H$ of generators)
such that $\tilde\sG$ becomes a Cayley graph, i.e.,
the group of isomorphisms of the coloured graph
is simply transitive on vertices.
Notice that $h \in H$ implies $h^{-1} \in H$.
Similarly, a {\sl Cayley structure} for a polytope
is a Cayley structure for its 1-skeleton.
Most semi-regular polytopes admit Cayley structures: in dimension 3,
only the dodecahedron and $\langle 3,5,3,5\rangle$ do not ([C1]).

Let $V_{\sG}$ be the complex vector space of functions
from the vertices of the Cayley graph $\sG$ to $\CC$.
Define $e_k$ by $e_k(g) = \delta_{kg}$.
The {\sl canonical left representation} of $\Gamma$ on $V_{\sG}$
is given by $L_g e_k = e_{gk}$ or,
equivalently, $(L_h v)(g) = v(h^{-1}g)$.
For a polytope with a Cayley structure,
the canonical left action corresponds
to an action by isometries.
The {\sl canonical right representation} is
given by $R_g e_k = e_{kg^{-1}}$.
The two canonical representations are equivalent, being
intertwined by the linear involution $e_k \mapsto e_{k^{-1}}$,
and commute: $L_{g_1} R_{g_2} = R_{g_2} L_{g_1}$.
Geometrically, this action can be interpreted in terms of
the Cayley graph $\sG$ as follows: 
for a generator $h$, the value of $R_h v$ at a vertex $g$
is the value of $v$ at the target vertex of the $h$-edge
starting from $g$ (which is, of course, $gh$),
i.e., $(R_h v)(g) = v(gh)$.
Thus, unless $\Gamma$ is abelian,
$R_h$ does not preserve adjacencies among vertices.

From representation theory for finite groups ([Se]),
the space $V_{\sG}$ splits as a direct sum,
$$V_{\sG} =
\bigoplus_{1 \le i \le t, 1 \le j \le d_i} W_{i,j},$$
where each $W_{i,j}$ is invariant under the canonical right action
and the restriction of the action to $W_{i,j}$
is isomorphic to the $i$-th irreducible representation.

As in Babai ([B]),
the adjacency matrix $X$ of the oriented graph $\sG$,
defined by $x_{ij} = 1$ if there is an (oriented) edge
from the $i$-th to the $j$-th vertex, and 0 otherwise, satisfies
$$X = \sum_{h \in H}R_h.$$
In particular, $X$ commutes with the left canonical action.
Invariant subspaces for the canonical right representation
are therefore invariant under $X$ and the restriction $X_{i,j}$ of
$X$ to each $W_{i,j}$ is the sum
of the restrictions of the linear transformations
corresponding to the elements of $H$.
Thus, if we have matrices $M_{h,i}$ for the generators in $H$
in any representation isomorphic to
the $i$-th irreducible representation of $\Gamma$,
$X_{i,j}$ is conjugate to
$$X_i = \sum_{h \in H}M_{h,i}.$$
Clearly, the spectrum $\sigma(X)$ of $X$ is the union (as a multiset)
of $d_i$ copies of $\sigma(X_i)$, $i = 1, \ldots, t$.

Although we use the right canonical action
to decompose $V_{\sG}$, the left canonical action
is geometrically more helpful in the identification
of a Cayley structure for a polytope.
Like Babai's method,
this technique applies to polytopes with different weights
for edges of different colours.

Instead of computing $X_i$,
Lov\'asz and Babai have a formula for traces of powers of $X_i$ ([B]):
$$\tr(X_i^\ell) =
\lambda_{i,1}^\ell + \cdots + \lambda_{i,{d_i}}^\ell =
\sum_{h_1,\ldots,h_\ell \in H}\chi_i(h_1\cdot\;\cdots\;\cdot h_\ell),$$
where $\lambda_{i,1},\ldots,\lambda_{i,{d_i}}$
are the eigenvalues of $X_i$, with dimensions $d_i$,
and $\chi_i$, $i = 1, \ldots, t$,
are the characters of the irreducible representations of $\Gamma$.
The traces of $X_i^\ell$ for
$\ell = 1, \ldots, d_i$, determine $\sigma(X_i)$.

In our examples, we find the explicit computation
of the right hand side of the above formula cumbersome
since it involves counting special paths
and we prefer to specify matrices for each irreducible representation.
