2021-09-12

Tags: #subjects/homological-stability #representation-theory

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

Reference: Church-Ellenberg-Farb

  • #open-problem : 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 \({\mathbb{Q}}[\kappa_i] \to H^*({ \mathcal{M}_{g, n} })\) for degree high enough.

attachments/2021-09-12_23-57-48.png

  • Main result: dimensions of representation stabilize.

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

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

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

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