  This paper studies the combinatoric structure of the set of all
representations, up to equivalence, of a finite-dimensional semisimple Lie
algebra. This has intrinsic interest as a previously unsolved problem in
representation theory, and also has applications to the understanding of
quantum decoherence. We prove that for Hilbert spaces of sufficiently high
dimension, decoherence-free subspaces exist for almost all representations of
the error algebra. For decoherence-free subsystems, we plot the function
$f_d(n)$ which is the fraction of all $d$-dimensional quantum systems which
preserve $n$ bits of information through DF subsystems, and note that this
function fits an inverse beta distribution. The mathematical tools which arise
include techniques from classical number theory.
