  Representation theory of finite groups portrays a marvelous crossroad of
group theory, algebraic combinatorics, and probability. In particular the
Plancherel measure is a probability that arises naturally from representation
theory, and in this thesis we consider three ramifications of asymptotic
questions for random Plancherel distributed representations. First we recall
irreducible characters of the symmetric group, which are indexed by integer
partitions. We focus on the so called 'dual approach', in which the partition
indexing a character is now considered to be the variable. We extend a famous
result of Kerov on the asymptotic of Plancherel distributed characters by
studying partial trace and partial sum of a representation matrix. We decompose
each of these objects into a main term and a remainder, and in each case we
prove a central limit theorem for the main term. We apply these results to
prove a law of large numbers for the partial sum. In the second part we
consider projective representations of the symmetric group by converting a
multirectangular approach developed by Stanley for the classical case. In
particular, we present a positivity conjecture for the coefficients of the
renormalized projective character, written as a polynomial in a new set of
coordinates. We prove the positivity for the leading term of the polynomial. In
the last part of the thesis we recall the theory of supercharacters, which is
an alternative tool to representation theory when the classical approach proves
to be intractable. We show that a generalization of the Plancherel measure,
called the superplancherel measure, arises naturally. We focus on a particular
supercharacter theory for the upper unitriangular group in which
supercharacters are indexed by set partitions. We prove a limit shape result
for a random set partition according to this distribution.
