2021-09-12

$${\mathsf{FI}}{\hbox{-}}$$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