  We study the local asymptotics at the edge for particle systems arising from:
(i) eigenvalues of sums of unitarily invariant random Hermitian matrices and
(ii) signatures corresponding to decompositions of tensor products of
representations of the unitary group. Our method treats these two models in
parallel, and is based on new formulas for observables described in terms of a
special family of lifts, which we call supersymmetric lifts, of Schur functions
and multivariate Bessel functions. We obtain explicit expressions for a class
of supersymmetric lifts inspired by determinantal formulas for supersymmetric
Schur functions due to \cite{MJ03}. Asymptotic analysis of these lifts enable
us to probe the edge. We focus on several settings where the Airy point process
arises.
