Tags: #homological_stability #representation_theory
\({\mathsf{FI}}{\hbox{-}}\)modules (23:45)
Reference: Church-Ellenberg-Farb
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#open_problems : what are the characters of representations for \(S_n\) acting on certain vector spaces:
- \(H^*({\mathrm{Conf}}_n(X))\)
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\(H^*({ \mathcal{M}_{g} })\) and its tautological ring \(R^*({ \mathcal{M}_{g} })\)
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Smallest subrings of [[Chow ring|Chow]] closed under pushforward by forgetful/gluing maps between various \({ \mathcal{M}_{g, n} }\).
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Can push through the cycle class map, unknown these are isomorphic.
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Can get a surjection \({\mathbb{Q}}[\kappa_i] \to H^*({ \mathcal{M}_{g, n} })\) for degree high enough.
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Main result: dimensions of representation stabilize.
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Sequence of \(S_n{\hbox{-}}\)reps converted into a single \({\mathsf{FI}}{\hbox{-}}\)module.
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\({\mathsf{{\mathsf{FI}}}{\hbox{-}}\mathsf{Mod}} \coloneqq F\in {\mathsf{Fun}}({\mathsf{FI}}, {\mathsf{k}{\hbox{-}}\mathsf{Mod}})\).
- Any \({\mathsf{FI}}{\hbox{-}}\)module provides a linear action \(\mathop{\mathrm{End}}_{{\mathsf{FI}}}(n) = S_n\).
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Gradings: functors from \({\mathbb{N}}\to {\mathsf{k}{\hbox{-}}\mathsf{Mod}}\).