  The degeneracy of energy levels in a quantum dot of Hall fluid, leading to
conductance peaks, can be readily derived from the partition functions of
conformal field theory. Their complete expressions can be found for Hall states
with both Abelian and non-Abelian statistics, upon adapting known results for
the annulus geometry. We analyze the Abelian states with hierarchical filling
fractions, \nu=m/(mp \pm 1), and find a non trivial pattern of conductance
peaks. In particular, each one of them occurs with a characteristic
multiplicity, that is due to the extended symmetry of the m-folded edge.
Experimental tests of the multiplicity can shed more light on the dynamics of
this composite edge.
