  English title: Eigenvalues of Symmetrized Shuffling Operators
  The random-to-random shuffling operator explains, for example, the evolution
of a deck of cards subject to the following random process: draw a card
randomly from the deck and reinsert it at a random position. If one instead
draws more than one card at a time before reinserting, then the resulting
operator is an example of a family of symmetrized shuffling operators studied
by Victor Reiner, Franco Saliola and Volkmar Welker.
  This thesis describes a way to obtain the eigenvalues of these operators. We
build on the work of Anton Dieker and Franco Saliola, who computed the
eigenvalues of the random-to-random shuffle. Here, we compute the eigenvalues
for all the operators of the family. We proceed with the help of the
representation theory of the symmetric group. We decompose the vector space on
which the shuffles act into simple modules for the symmetric group. These
modules correspond to standard Young tableaux, and the algorithm to compute the
eigenvalues is combinatorial because it computes the eigenvalues directly from
the standard Young tableaux.
  As a corollary of our main result, we solve several conjectures of Reiner,
Saliola and Welker, including showing that the eigenvalues are all nonnegative
integers. Furthermore, the techniques used here allow us to give a new proof of
their result that these symmetrized shuffling operators commute. Knowing the
eigenvalues is the key step in one method of computing the number of shuffles
one needs to execute to get a perfectly shuffled deck, which is briefly
explored.
  We also study a second family of shuffles introduced by Reiner, Saliola and
Welker. We present many conjectures about their eigenvalues.
