---
date: 2021-09-12
tags: [ web/quick-notes ]
---


# 2021-09-12

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

## $\FI\dash$modules (23:45)

> Reference: [Church-Ellenberg-Farb](https://arxiv.org/pdf/1204.4533.pdf)

- #open/problems : what are the [characters](characters.md) of representations for $S_n$ acting on certain vector spaces:
  - $H^*(\Conf_n(X))$
  - $H^*(\mg)$ and its tautological ring $R^*(\mg)$
    - Smallest subrings of [Chow](Chow%20ring.md) closed under [pushforward](pushforward.md) by forgetful/gluing maps between various $\mgn$.
    
	- Can push through the [cycle class map](cycle%20class%20map), unknown these are isomorphic.
    
	- Can get a surjection $\QQ[\kappa_i] \to H^*(\mgn)$ for degree high enough.

![](attachments/2021-09-12_23-57-48.png)

- Main result: dimensions of representation stabilize.
- Sequence of $S_n\dash$reps converted into a single $\FI\dash$module.

- $\mods{\FI} \da F\in \Fun(\FI, \mods{k})$.
  - Any $\FI\dash$module provides a linear action $\Endo_{\FI}(n) = S_n$.

- Gradings: functors from $\NN\to \mods{k}$.