We study a class of representations of the Cuntz algebras O_N, N=2,3,...,
acting on L^2(T) where T=R/2\pi Z. The representations arise in wavelet theory,
but are of independent interest. We find and describe the decomposition into
irreducibles, and show how the O_N-irreducibles decompose when restricted to
the subalgebra UHF_N\subset O_N of gauge-invariant elements; and we show that
the whole structure is accounted for by arithmetic and combinatorial properties
of the integers Z. We have general results on a class of representations of O_N
on Hilbert space H such that the generators S_i as operators permute the
elements in some orthonormal basis for H. We then use this to extend our
results from L^2(T) to L^2(T^d), d>1 ; even to L^2(\mathbf{T}) where \mathbf{T}
is some fractal version of the torus which carries more of the algebraic
information encoded in our representations.Comment: 84 pages, 11 figures, AMS-LaTeX v1.2b, full-resolution figures
available at ftp://ftp.math.uiowa.edu/pub/jorgen/PermRepCuntzAlg in eps files
with the same names as the low-resolution figures included her