• stability patterns
• representation stability

The topic of homological stability is well-known and deeply studied.
Classically, given a sequence of groups or topological spaces {X_n} equipped with maps φ_n : X_n->X_{n+1}, we call this sequence homologically stable over a coefficient ring R if for each k positive integer the map
(φ_n)_* : H_k(X_n;R)->H_k(X_{n+1};R) is an isomorphism for all n greater than a certain stable range n_0, possibly depending on k.

The article by Church and Farb, "Representation theory and homological
stability", offers a new viewpoint, giving us the chance to study a different concept of stability, using instruments belonging to representation theory.
We can give different definitions, more or less strong, of representation stability, analyzing families of groups {G_n} of type S_n, GL_n(Q), SL_n(Q) or Sp_{2n}(Q), for which we have natural injective maps G_n -> G_{n+1}. The irreducible representations of G_n are classified, respectively, throughout partitions, pseudo-partitions or double partitions and we will universally denote them as V(λ)_n, where λ will denote a partition, a pseudo-partition or a double partition.
Given a sequence of groups {G_n} of one of the indicated type and a G_n-representation V_n for all n, with maps φ_n : V_n->V_{n+1}, we say that
the sequence {V_n} is consistent if the action of G_n “commutes” with φ_n.
A consistent sequence of G_n-representation is representation stable if there exists a stable range n_0 such that for all n greater or equal than n_0
1. φ_n : V_n->V_{n+1} is injective;
2. the G_{n+1}-span of φ_n(V_n) is V_{n+1};
3. given the decomposition in irreducible components of a representation V_n, the multiplicities of the irreducibles are eventually indipendent of n.
If the stable range n_0 is indipendent of λ, we speak about uniform representation stability. Moreover, we can give a further refinement of this definition if we ask for a condition of “type-preserving”: given a subrepresentation U of V_n such that U is isomorphic to the irreducible representation V(λ)_n, the space generated by the orbit of φ_n(U) under the action of G_{n+1} is isomorphic to V(λ)_{n+1}. In this case we speak about strong stability.

After the general introduction to representation stability, we will provide, accordingly to "Stability patterns in Representation Theory" by Sam and Snowden, a categorical framework in which we will be able to classify irreducible objects in the category of representations of the “limit group” G defined as the union of all G_n, developing this exposition in particular for the symmetric group and the general linear group.

Once we have introduced the general theory, we will turn our attention
to some particular examples, as the (uniform) representation stability
for the cohomology of the pure braid group, giving, also thanks to the well
known characterization explained by Arnol’d in "The cohomology ring of the
coloured braid group", an explicit calculation of the decomposition in irreducible representations for the first cohomology groups, observing how the direct computation becomes fastly quite complicated.

To give another application, with the aim of clarify the potentiality of this new approach for the study of stability phenomena, we will also examine the representation stability for Lie algebras. In this case, in particular, we will focus on the specific case of free Lie algebras.