  Globular Clusters (GCs) provide valuable insight into the properties of their
host galaxies' dark matter halos. Using N-body simulations incorporating
semianalytic dynamical friction and GC-GC merger prescriptions, we study the
evolution of GC radial distributions and mass functions in cuspy and cored dark
matter halos. Modeling the dynamics of the GC-rich system in the dwarf galaxy
UGC 7369, we find that friction-induced inspiral and subsequent mergers of
massive GCs can naturally and robustly explain the mass segregation of the GCs
and the existence of a nuclear star cluster (NSC). However, the multiple
mergers required to form the NSC only take place when the dark matter halo is
cuspy. In a cored halo, stalling of the dynamical friction within the core
halts the inspiral of the GCs, and so the GC merger rate falls significantly,
precluding the formation of an NSC. We therefore argue that the presence of an
NSC requires a cusp in UGC 7369. More generally, we propose that the presence
of an NSC and the corresponding alteration of the GC mass function due to
mergers may be used as an indicator of a cuspy halo for galaxies in which we
expect NSC formation to be merger-dominated. These observables represent a
simple, powerful complement to other inner halo density profile constraint
techniques, and should allow for straightforward extension to larger samples.
