  We calculate the dominant excitations for the $k$-level ($k\in\mathbb{N}$)
Read-Rezayi (RR) states and their particle-hole conjugates, the anti
Read-Rezayi ($\bar{\textrm{RR}}$), proposed for quantum Hall states. These
states are supposed to be build over the second Landau level with total filling
factor $\nu=2+\nu^*$ with $\nu^*=k/(k+2)$ for RR and $\nu^*=2/(k+2)$ for
$\bar{\textrm{RR}}$. In the $k$-level RR states, based on $\mathbb{Z}_k$
parafermions, the dominant excitations are the fundamental quasiparticles with
fractional charge $e^*_k= e/(k + 2)$, with $e$ the electron charge, if $k=2,3$.
For k=4 the single-qp and the 2-agglomerate, with charge $2e^*_k$, have the
same scaling and both dominate, while for $k>4$ the 2-agglomerates are
dominant. Anyway the dominance of the 2-agglomerates can be affected by the
presence of environmental renormalizations. For all the $k$-level
$\bar{\textrm{RR}}$ states the single-qp and the 2-agglomerate have the same
scaling and both dominate. In this case only the presence of environmental
renormalizations can make one dominant over the other. We determine the
conditions where the environmental renormalizations of the charged and neutral
modes make dominant the Abelian 2-agglomerates over the non-Abelian
single-quasiparticles in the two models and for any value of $k$. We conclude
observing that, according these predictions, the dominance of 2-agglomerates,
at very low energies for the $\nu=5/2$, can be an interesting indication
supporting the validity of the anti-Pfaffian model in comparison to the
Pfaffian.
