Tags: #homotopy/homological-stability #lie-theory

\({\mathsf{FI}}{\hbox{-}}\)modules (23:45)

Reference: Church-Ellenberg-Farb

  • #open/problems : what are the characters of representations for \(S_n\) acting on certain vector spaces:
    • \(H^*({\mathrm{Conf}}_n(X))\)
    • \(H^*({ \mathcal{M}_{g} })\) and its tautological ring \(R^*({ \mathcal{M}_{g} })\)
      • Smallest subrings of Chow closed under pushforward by forgetful/gluing maps between various \({ \mathcal{M}_{g, n} }\).

      • Can push through the cycle class map, unknown these are isomorphic.

      • Can get a surjection \({\mathbf{Q}}[\kappa_i] \to H^*({ \mathcal{M}_{g, n} })\) for degree high enough.


  • Main result: dimensions of representation stabilize.

  • Sequence of \(S_n{\hbox{-}}\)reps converted into a single \({\mathsf{FI}}{\hbox{-}}\)module.

  • \({}_{{\mathsf{FI}}}{\mathsf{Mod}} \coloneqq F\in {\mathsf{Fun}}({\mathsf{FI}}, {}_{k}{\mathsf{Mod}} )\).

    • Any \({\mathsf{FI}}{\hbox{-}}\)module provides a linear action \({ \operatorname{End} }_{{\mathsf{FI}}}(n) = S_n\).
  • Gradings: functors from ${\mathbb{N}}\to {}_{k}{\mathsf{Mod}} $.

#web/quick-notes #homotopy/homological-stability #lie-theory #open/problems