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 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.
