The question title basically has my question. What is the precise definition/construction of$$\text{Sym}(\text{Sym}^2),$$as mentioned in the following?


<http://aimpl.org/repnstability/1/>


If possible, I would like this to be self-contained and not defer to other stuff about twisted commutative algebras, as much as they are interesting.



Precise definition/construction of $\text{Sym}(\text{Sym}^2)$?
As Dan Petersen points out, it's impossible to answer this question without talking about twisted commutative algebras. Moreover, to talk about TCAs over $\mathbb{Z}$, you need a more general definition of TCAs (which becomes equivalent to the earlier Sam-Snowden definitions when you're over $\mathbb{C}$). I'm not sure this is in the literature, so I'll try to explain.


**Definition:** A TCA is a commutative algebra inside the tensor category of FB-modules.


Let's expand this definition. FB denotes the category of finite sets and bijections, and an FB-module is a functor $W: \mathrm{FB}\to \mathbb{Z}\mathrm{-Mod}$. In other words, $W$ is essentially a collection of $S\_n$-representations $W\_n$ (one for each $n\in \mathbb{N}$), with no relation between them.


The disjoint union of finite sets $\sqcup:\mathrm{FB}\times \mathrm{FB}\to \mathrm{FB}$ induces a tensor product $\otimes:\mathrm{FB}\mathrm{-Mod}\times \mathrm{FB}\mathrm{-Mod}\to \mathrm{FB}\mathrm{-Mod}$ (often called the "convolution tensor product". It has the universal property that a map of FB-modules $A\otimes B\to C$ is equivalent to a collection of maps $A\_S\otimes\_{\mathbb{Z}}B\_T\to C\_{S\sqcup T}$ for all finite sets $S$ and $T$, natural in $S$ and $T$. (Equivalently, you can say it's a collection of $S\_k\times S\_\ell$-equivariant maps $A\_k\otimes\_{\mathbb{Z}}B\_\ell\to C\_{k+\ell}$, but it's easier to keep track of things like the symmetry $A\otimes B\cong B\otimes A$ with the former definition.)


A TCA is an FB-module $A$ together with a multiplication $A\otimes A\to A$ (which has to be commutative, but this is a property not structure so we don't need to stress it).




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Now we can define $\mathrm{Sym}(\mathrm{Sym}^2)$. Recall that a perfect matching on a set $T$ is an equivalence relation on $T$ where each equivalence class has exactly 2 elements.


**Definition:** $\mathrm{Sym}(\mathrm{Sym}^2)$ is the FB-module $A$ which assigns to a finite set $T$ the free abelian group $A\_T=\mathbb{Z}[\{\text{perfect matchings on }T\}]$.


To make this a TCA, we need to define the multiplication map $A\otimes A\to A$. According to the universal property above, this means we have to specify natural maps $A\_S\otimes\_{\mathbb{Z}} A\_T\to A\_{S\sqcup T}$. But this is very easy: if you have a perfect matching of $S$, and a perfect matching of $T$, their union is a perfect matching of $S\sqcup T$.




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From this perspective the name may not be so clear. Here are two explanations. First (and perhaps less satisfying for you?): Define an FB-module $X$ by $X\_T=\begin{cases}\mathbb{Z}&\text{if }\lvert T\rvert=1\\0&\text{else}\end{cases}$. Because we have a tensor product, we can make sense of $\textrm{Sym}^k$ in FB-Mod, and for example $\textrm{Sym}^2 X$ turns out to be $(\textrm{Sym}^2 X)\_T=\begin{cases}\mathbb{Z}&\text{if }\lvert T\rvert=2\\0&\text{else}\end{cases}$. Moreover, our algebra $A$ is precisely $\textrm{Sym}^\*(\textrm{Sym}^2 X)=\bigoplus\_{k\in \mathbb{N}}\textrm{Sym}^k(\textrm{Sym}^2 X)$.


Alternately, if you were to work over $\mathbb{C}$ throughout, then there is an alternate description of TCAs over $\mathbb{C}$ in terms of polynomial representations of $\textrm{GL}\_\infty(\mathbb{C})$. (See e.g. Section 2.1 of [arXiv:1501.06925](https://arxiv.org/abs/1501.06925v2).) In this perspective, you are looking at nothing other than the commutative ring $\textrm{Sym}^\*(\textrm{Sym}^2 \mathbb{C}^\infty)$, with its natural multiplication and natural $\textrm{GL}\_\infty(\mathbb{C})$-action.


