representationstables
Representationstables refers to a concept in representation theory describing sequences of representations that become stable as the underlying parameter grows. Typically, one considers a family of groups G_n (often the symmetric groups S_n) and a sequence of G_n-modules V_n together with structure maps that are compatible with the G_n-actions. A representationstable sequence exhibits that, for large n, the decomposition of V_n into irreducible components stabilizes: both the types of irreducibles and their multiplicities stop changing in a predictable way. In many settings, dimensions grow polynomially and characters of V_n become governed by fixed polynomials.
A common framework for studying representationstables is the theory of FI-modules. An FI-module is a functor
Typical examples include the cohomology groups of configuration spaces, H^i(Conf_n(R^d)), viewed as S_n-representations, which become stable
Applications of representationstables span algebraic topology, algebraic geometry, combinatorics, and representation theory, providing unified patterns for