It is known that there is a representation of the affine Lie algebra $\widehat{\mathfrak{sl}\_q}$ (over $\mathbb{Z}$) on the algebra of symmetric functions, where the action of the Chevalley generators $E\_i,F\_i$, $i=0, \ldots q-1$ on the basis of Schur polynomials $s\_\lambda$ is given by
$$E\_i s\_\lambda= \sum\_{\lambda\setminus \mu \text{ is an } i-\text{cell}} s\_\mu$$
$$F\_i s\_\lambda= \sum\_{\mu\setminus\lambda \text{ is an } i-\text{cell}} s\_\mu$$
where a cell $(x,y)$ in Young diagram is called an $i$-cell if $x-y\equiv i\pmod{q}$.


Using the fact that this action commutes with multiplication by the power sum symmetric functions $p\_{qk}$ for all $k\in \mathbb{Z}\_{\ge 1}$, one can show that the $\mathbb{Q}$-vector space generated by the action of $\widehat{\mathfrak{sl}\_q}\otimes \mathbb{Q}$ on $s\_\emptyset$ is $\mathbb{Q}\left[p\_n\ |\ q\nmid n\right]$.


I am interested, for $q$ prime, in the $\mathbb{Z}\_{(q)}$-lattice generated by the action of $\widehat{\mathfrak{sl}\_q}\otimes \mathbb{Z}\_{(q)}$ on $s\_\emptyset$, where $\mathbb{Z}\_{(q)}$ is the subring of $\mathbb{Q}$ consisting of fractions $\frac{r}{s}$ with $s$ not divisible by $q$. From Darij Grinberg's [answer](https://mathoverflow.net/a/403672/160416) to a previous question of mine, one can see that this is contained in $\mathbb{Z}\_{(q)}\left[p\_r\ |\ q\nmid r\right]$. Computations suggest that equality should hold for $q>2$, but for $q=2$ the situation is different.


I observed that the $\mathbb{Z}\_{(2)}$-lattice generated by the action of $\widehat{\mathfrak{sl}\_2}\otimes \mathbb{Z}\_{(2)}$ on $s\_\emptyset$ seems to be $\mathbb{Z}\_{(2)}\left[p\_1,2p\_3,4p\_5,8p\_7, \ldots\right]$. It's not clear to me why this particular lattice should be stable under the operators $E\_0,E\_1,F\_0,F\_1$ and why the prime $2$ behaves differently here. I am therefore looking for an explanation of this phenomenon.


I am not very familiar with affine Lie algebras and their representations, so please excuse me if I said anything wrong or if this follows easily from well-known facts about highest-weight modules.



Action of $\widehat{\mathfrak{sl}_2}$ on symmetric functions with $\mathbb{Z}_{(2)}$ coefficients
I ended up finding a proof of this phenomenon using the vertex operator description of the basic representation of $\widehat{\mathfrak{sl\_2}}$. It can be found in the following preprint: <https://arxiv.org/abs/2211.10584>.


