Given a ring spectrum $R$ and an $R$-module $E$, we have the spectral symmetric algebra $\mathrm{Sym}_R(E)$ of $E$ over $R$, defined by $$ \begin{align*} \mathrm{Sym}_R(E) &\overset{\mathrm{def}}{=} \mathrm{colim}_{\mathbb{F}}(\Delta_{E})\\ &\cong \bigoplus_{n\in\mathbb{N}}E^{\otimes_\mathbb{S}n}_{\mathsf{h}\Sigma_{n}}, \end{align*} $$ where $\mathbb{F}\overset{\mathrm{def}}{=}\mathsf{FinSets}^\simeq$ is the groupoid of finite sets and permutations. As A Rock and a Hard Place showed here, the $\mathbb{E}_\infty$-ring $\mathrm{Sym}_R(E)$ comes with a natural grading by the sphere spectrum, inducing a $\mathbb{Z}$-grading on $\pi_0(\mathrm{Sym}_R(E))\cong\mathrm{Sym}_{\pi_0(R)}(\pi_0(E))$. So e.g. picking $R=E=\mathbb{S}$, gives $$ \begin{align*} \pi_0(\mathrm{Sym}_{\mathbb{S}}(\mathbb{S})) &\overset{\mathrm{def}}{=} \pi_0(\mathbb{S}\{t\})\\ &\cong \mathbb{Z}[t], \end{align*} $$ which carries the natural $\mathbb{Z}$-grading.

However, the $\pi_0$ of an $\mathbb{S}$-graded ring can be more complicated than just a commutative $\mathbb{Z}$-grading, and for instance allows for the multiplication on the $\pi_0$ to be supercommutative, satisfying $ab=(-1)^{\deg(a)\deg(b)}ba$. This led me to the following pair of questions:

- Is there an "spectral exterior algebra" construction $\bigwedge_RE$, whose $\pi_0$ is the $\mathbb{Z}$-graded supercommutative exterior algebra $\bigwedge_{\pi_0(R)}\pi_0(E)$? If so, does it come with an $\mathbb{S}$-grading?
- One of the more homotopy-theoretic points of view on symmetric and exterior algebras is that the passage from the former to the latter corresponds to considering a larger portion of the sphere spectrum. More generally, do we have an $\mathbb{N}$-indexed sequence of "higher exterior algebra" constructions $\mathrm{Sym}_R(E)$, $\bigwedge_R(E)$, $\bigwedge^{\mathbf{2}}_R(E)$, $\ldots$?