Let $S_n$ denote the symmetric group on $n$-letters and let $\mathrm{Rep}(S_n)$ denote its representation ring. For every $n$ restriction along the inclusion $S_{n-1} \to S_n$ induces a ring homomorphism $\mathrm{Res}^{S_n}_{S_{n-1}} :\mathrm{Rep}(S_n) \to \mathrm{Rep}(S_{n-1 })$. Considering these together we get a diagram of the following form $$\dots \to \mathrm{Rep}(S_n) \to \mathrm{Rep}(S_{n-1 }) \to \dots \to \mathrm{Rep}(S_2) \to \mathrm{Rep}(S_1) \simeq \mathbb{Z}$$

Question:Is there a generators and relations description of the limit $\underset{n }{\varprojlim} \,\mathrm{Rep}(S_n)$ as ring?

**Edit:** Will Sawin has pointed out in the comments that the limit, must have uncountably many generators in any presentation. Let me slightly modify the question in the hopes of making it more answerable.

Lets consider instead of $\mathrm{Rep}(S_n)$ its completion with respect to the augmentation ideal $\widehat{\mathrm{Rep}}(S_n):=\varprojlim_m \mathrm{Rep}(S_n)/I^m$ where $I:= \ker(\dim : \mathrm{Rep}(S_n) \to \mathbb{Z})$. We still have a tower as above and we can ask the following variation.

Question:Is there an explicit countable presentation of the limit $\underset{n }{\varprojlim} \,\widehat{\mathrm{Rep}}(S_n)$ in some suitable category of complete topological rings?

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