  In the first part of this thesis, we make use of representation theory of
groups and algebras to perform an irreducible decomposition of tensors in the
context of metric-affine gravity. In particular, we consider the action of the
orthogonal group O(1, d$-1$) on the Riemann tensor associated with an affine
connection defined on a d-dimensional pseudo-Riemannian manifold. This
connection, with torsion and non-metricity, is the characteristic ingredient of
metric-affine theories of gravity.
  In the second part of this thesis, we construct the projection operators used
for the aforementioned decomposition. They are realized in terms of the
symmetric group algebra $\mathbb{C}\mathfrak{S}_n$ and of the Brauer algebra
B$_n$(d) which are related respectively to the action of GL(d,$\mathbb{C}$)
(and its real form GL(d,$\mathbb{R}$)) and to the action of O(d,$\mathbb{C}$)
(and its real form O(1 , d$-1$)) on tensors via the Schur-Weyl duality. First
of all, we give an alternative approach to the known formulas for the central
idempotents of $\mathbb{C}\mathfrak{S}_n$. These elements provide a unique
reducible decomposition, known as the isotypic decomposition. For our purposes,
this decomposition is remarkably handy to arrive at the sought after
irreducible decomposition with respect to GL(d,$\mathbb{R}$). Then, we
construct the elements in B$_n$(d) which realize the isotypic decomposition of
a tensor under the action of O(d,$\mathbb{C}$). This decomposition is
irreducible under O(d,$\mathbb{C}$) when applied to an irreducible
GL(d,$\mathbb{C}$) tensor of order $5$ or less. As a by product of the
construction, we give a solution to the problem of decomposing an arbitrary
tensor into its traceless part, doubly traceless part and so on. These results
led to the development of several Mathematica packages linked to the
\textit{xAct} bundle for tensor calculus in field theory.
