354 AUDREY TERRAS, ARITHMETICAL QUANTUM CHAOS
M. Minei [60] has noticed that one can draw very different pictures of the spec-
tra (see Hofstadter's butterfly in Figure 12) for these graphs by using Hofstadter's
idea which leads to separating the spectra corresponding to the q-dimensional rep-
resentations of the Heisenberg group H('ll/q'll). We defined these representations
'lrr, r = 1, ... , q - 1 in [24]. Plot the part of the spectrum of the Cayley graph
corresponding to 7r r as points in the plane with y-coordinate ~ and x-coordinates
given by the eigenvalues A of the matrix
Mr= L 7rr(s).
sES
Of course A must lie in the interval [-4, 4]. Hofstadter was interested in matrices
analogous to those from graphs for the Heisenberg group over '1l itself which is the
limiting picture.
heis mod 132 of degree 40.8
0 .7
0.6
0.5
0.4
0.3
0 .2Figure 12. Hofstadter's Butterfly for the Highest Dimensiona l Part of the
Spectrum of the Cayley Graph H(169). The figure is obtained as d escribed in
the text.The moral of Figure 12 is that there could be more useful ways of representing
spectra than just histograms. But one needs a certain structure of the representa-
tions of the symmetry group in order to be able to see butterflies.