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As Peter points out, it is more reasonable to look for connections between prisms and homotopy theory. I'll answer my own queston with some recent observations/speculations on the relation between prisms and equivariant homotopy theory, most of which are recorded in my paper A slice refinement of Bökstedt periodicity.

• Let $A=\mathbb Z[q]$, acted on by Adams operations $\psi^k(q)=q^k$, and write $B^\bullet$ for the multiplicative monoid underlying a semiring $B$. Then $n \mapsto ([n]_q, \psi^n)$ defines a monoid map $$\operatorname{End}({\mathbb T}) \ge \mathbb N^\bullet \to A^\bullet \rtimes \operatorname{End}_{\mathrm{Ring}}(A).$$ For an oriented prism $(A,d)$ we can do something similar with the submonoid $p^{\mathbb N}\le\mathbb N^\bullet$ by sending $p^k\mapsto (d\dotsm\phi^{k-1}(d), \phi^k)$.

• The prism condition $\delta(d)\in A^\times$ is equivalent to $FV=p$ (Corollary 3.28), which is a special case of the Mackey functor condition $\operatorname{res}(\operatorname{tr}(x)) = \sum_{g\in G/H} g\cdot x$.

• $q$-divided powers give lifts of the Norm map (Propositions 3.32 and 3.33), which is part of the Tambara functor structure of $\mathrm{THH}$. Conversely, the fact that $q$-divided powers descend to a map $W_1(R) \to W_2(R)$ is essentially equivalent to the "existence of higher $q$-divided powers" lemma (Lemma 16.7 of Prisms and prismatic cohomology)

• In view of the previous point, the fact that the Norm scales slice filtration is closely related to a lemma needed for the convergence of the $q$-logarithm (Remark 1.4 of my paper and Proposition 4.9 of The $p$-completed cyclotomic trace in degree 2)

So this suggests it might be possible to define a notion of $G$-prism for a compact Lie group $G$, recovering prisms for $G=\mathbb Q_p/\mathbb Z_p$, presumably related to Dress-Siebeneicher's $G$-Witt vectors, such that $\pi^G_0$ of a nice $G$-$E_\infty$-ring spectrum is a $G$-prism.

Edit Sep 2023: the ideas below have now been developed more thoroughly in my new paper Prisms and Tambara functors I

As Peter points out, it is more reasonable to look for connections between prisms and homotopy theory. I'll answer my own queston with some recent observations/speculations on the relation between prisms and equivariant homotopy theory, most of which are recorded in my paper A slice refinement of Bökstedt periodicity.

• Let $A=\mathbb Z[q]$, acted on by Adams operations $\psi^k(q)=q^k$, and write $B^\bullet$ for the multiplicative monoid underlying a semiring $B$. Then $n \mapsto ([n]_q, \psi^n)$ defines a monoid map $$\operatorname{End}({\mathbb T}) \ge \mathbb N^\bullet \to A^\bullet \rtimes \operatorname{End}_{\mathrm{Ring}}(A).$$ For an oriented prism $(A,d)$ we can do something similar with the submonoid $p^{\mathbb N}\le\mathbb N^\bullet$ by sending $p^k\mapsto (d\dotsm\phi^{k-1}(d), \phi^k)$.

• The prism condition $\delta(d)\in A^\times$ is equivalent to $FV=p$ (Corollary 3.28), which is a special case of the Mackey functor condition $\operatorname{res}(\operatorname{tr}(x)) = \sum_{g\in G/H} g\cdot x$.

• $q$-divided powers give lifts of the Norm map (Propositions 3.32 and 3.33), which is part of the Tambara functor structure of $\mathrm{THH}$. Conversely, the fact that $q$-divided powers descend to a map $W_1(R) \to W_2(R)$ is essentially equivalent to the "existence of higher $q$-divided powers" lemma (Lemma 16.7 of Prisms and prismatic cohomology)

• In view of the previous point, the fact that the Norm scales slice filtration is closely related to a lemma needed for the convergence of the $q$-logarithm (Remark 1.4 of my paper and Proposition 4.9 of The $p$-completed cyclotomic trace in degree 2)

So this suggests it might be possible to define a notion of $G$-prism for a compact Lie group $G$, recovering prisms for $G=\mathbb Q_p/\mathbb Z_p$, presumably related to Dress-Siebeneicher's $G$-Witt vectors, such that $\pi^G_0$ of a nice $G$-$E_\infty$-ring spectrum is a $G$-prism.

As Peter points out, it is more reasonable to look for connections between prisms and homotopy theory. I'll answer my own queston with some recent observations/speculations on the relation between prisms and equivariant homotopy theory, most of which are recorded in my paper A slice refinement of Bökstedt periodicity.

• Let $A=\mathbb Z[q]$, acted on by Adams operations $\psi^k(q)=q^k$, and write $B^\bullet$ for the multiplicative monoid underlying a semiring $B$. Then $n \mapsto ([n]_q, \psi^n)$ defines a monoid map $$\operatorname{End}({\mathbb T}) \ge \mathbb N^\bullet \to A^\bullet \rtimes \operatorname{End}_{\mathrm{Ring}}(A).$$ For an oriented prism $(A,d)$ we can do something similar with the submonoid $p^{\mathbb N}\le\mathbb N^\bullet$ by sending $p^k\mapsto (d\dotsm\phi^{k-1}(d), \phi^k)$.

• The prism condition $\delta(d)\in A^\times$ is equivalent to $FV=p$ (Corollary 3.28), which is a special case of the Mackey functor condition $\operatorname{res}(\operatorname{tr}(x)) = \sum_{g\in G} g\cdot x$.

• $q$-divided powers give lifts of the Norm map (Propositions 3.32 and 3.33), which is part of the Tambara functor structure of $\mathrm{THH}$. Conversely, the fact that $q$-divided powers descend to a map $W_1(R) \to W_2(R)$ is essentially equivalent to the "existence of higher $q$-divided powers" lemma (Lemma 16.7 of Prisms and prismatic cohomology)

• In view of the previous point, the fact that the Norm scales slice filtration is closely related to a lemma needed for the convergence of the $q$-logarithm (Remark 1.4 of my paper and Proposition 4.9 of The $p$-completed cyclotomic trace in degree 2)

So this suggests it might be possible to define a notion of $G$-prism for a compact Lie group $G$, recovering prisms for $G=\mathbb Q_p/\mathbb Z_p$, presumably related to Dress-Siebeneicher's $G$-Witt vectors, such that $\pi^G_0$ of a nice $G$-$E_\infty$-ring spectrum is a $G$-prism.

As Peter points out, it is more reasonable to look for connections between prisms and homotopy theory. I'll answer my own queston with some recent observations/speculations on the relation between prisms and equivariant homotopy theory, most of which are recorded in my paper A slice refinement of Bökstedt periodicity.

• Let $A=\mathbb Z[q]$, acted on by Adams operations $\psi^k(q)=q^k$, and write $B^\bullet$ for the multiplicative monoid underlying a semiring $B$. Then $n \mapsto ([n]_q, \psi^n)$ defines a monoid map $$\operatorname{End}({\mathbb T}) \ge \mathbb N^\bullet \to A^\bullet \rtimes \operatorname{End}_{\mathrm{Ring}}(A).$$ For an oriented prism $(A,d)$ we can do something similar with the submonoid $p^{\mathbb N}\le\mathbb N^\bullet$ by sending $p^k\mapsto (d\dotsm\phi^{k-1}(d), \phi^k)$.

• The prism condition $\delta(d)\in A^\times$ is equivalent to $FV=p$ (Corollary 3.28), which is a special case of the Mackey functor condition $\operatorname{res}(\operatorname{tr}(x)) = \sum_{g\in G/H} g\cdot x$.

• $q$-divided powers give lifts of the Norm map (Propositions 3.32 and 3.33), which is part of the Tambara functor structure of $\mathrm{THH}$. Conversely, the fact that $q$-divided powers descend to a map $W_1(R) \to W_2(R)$ is essentially equivalent to the "existence of higher $q$-divided powers" lemma (Lemma 16.7 of Prisms and prismatic cohomology)

• In view of the previous point, the fact that the Norm scales slice filtration is closely related to a lemma needed for the convergence of the $q$-logarithm (Remark 1.4 of my paper and Proposition 4.9 of The $p$-completed cyclotomic trace in degree 2)

So this suggests it might be possible to define a notion of $G$-prism for a compact Lie group $G$, recovering prisms for $G=\mathbb Q_p/\mathbb Z_p$, presumably related to Dress-Siebeneicher's $G$-Witt vectors, such that $\pi^G_0$ of a nice $G$-$E_\infty$-ring spectrum is a $G$-prism.

As Peter points out, it is more reasonable to look for connections between prisms and homotopy theory. I'll answer my own queston with some recent observations/speculations on the relation between prisms and equivariant homotopy theory, most of which are recorded in my paper A slice refinement of Bökstedt periodicity.

• Let $A=\mathbb Z[q]$, acted on by Adams operations $\psi^k(q)=q^k$, and write $B^\bullet$ for the multiplicative monoid underlying a semiring $B$. Then $n \mapsto ([n]_q, \psi^n)$ defines a monoid map $$\operatorname{End}({\mathbb T}) \supset \mathbb N^\bullet \to A^\bullet \rtimes \operatorname{End}_{\mathrm{Ring}}(A).$$ For an oriented prism $(A,d)$ we can do something similar with the submonoid $p^{\mathbb N}\le\mathbb N^\bullet$ by sending $p^k\mapsto (d\dotsm\phi^{k-1}(d), \phi^k)$.

• The prism condition $\delta(d)\in A^\times$ is equivalent to $FV=p$ (Corollary 3.28), which is a special case of the Mackey functor condition $\operatorname{res}(\operatorname{tr}(x)) = \sum_{g\in G} g\cdot x$.

• $q$-divided powers give lifts of the Norm map (Propositions 3.32 and 3.33), which is part of the Tambara functor structure of $\mathrm{THH}$. Conversely, the fact that $q$-divided powers descend to a map $W_1(R) \to W_2(R)$ is essentially equivalent to the "existence of higher $q$-divided powers" lemma (Lemma 16.7 of Prisms and prismatic cohomology)

• In view of the previous point, the fact that the Norm scales slice filtration is closely related to a lemma needed for the convergence of the $q$-logarithm (Remark 1.4 of my paper and Proposition 4.9 of The $p$-completed cyclotomic trace in degree 2)

So this suggests it might be possible to define a notion of $G$-prism for a compact Lie group $G$, recovering prisms for $G=\mathbb Q_p/\mathbb Z_p$, presumably related to Dress-Siebeneicher's $G$-Witt vectors, such that $\pi^G_0$ of a nice $G$-$E_\infty$-ring spectrum is a $G$-prism.

As Peter points out, it is more reasonable to look for connections between prisms and homotopy theory. I'll answer my own queston with some recent observations/speculations on the relation between prisms and equivariant homotopy theory, most of which are recorded in my paper A slice refinement of Bökstedt periodicity.

• Let $A=\mathbb Z[q]$, acted on by Adams operations $\psi^k(q)=q^k$, and write $B^\bullet$ for the multiplicative monoid underlying a semiring $B$. Then $n \mapsto ([n]_q, \psi^n)$ defines a monoid map $$\operatorname{End}({\mathbb T}) \ge \mathbb N^\bullet \to A^\bullet \rtimes \operatorname{End}_{\mathrm{Ring}}(A).$$ For an oriented prism $(A,d)$ we can do something similar with the submonoid $p^{\mathbb N}\le\mathbb N^\bullet$ by sending $p^k\mapsto (d\dotsm\phi^{k-1}(d), \phi^k)$.

• The prism condition $\delta(d)\in A^\times$ is equivalent to $FV=p$ (Corollary 3.28), which is a special case of the Mackey functor condition $\operatorname{res}(\operatorname{tr}(x)) = \sum_{g\in G} g\cdot x$.

• $q$-divided powers give lifts of the Norm map (Propositions 3.32 and 3.33), which is part of the Tambara functor structure of $\mathrm{THH}$. Conversely, the fact that $q$-divided powers descend to a map $W_1(R) \to W_2(R)$ is essentially equivalent to the "existence of higher $q$-divided powers" lemma (Lemma 16.7 of Prisms and prismatic cohomology)

• In view of the previous point, the fact that the Norm scales slice filtration is closely related to a lemma needed for the convergence of the $q$-logarithm (Remark 1.4 of my paper and Proposition 4.9 of The $p$-completed cyclotomic trace in degree 2)

So this suggests it might be possible to define a notion of $G$-prism for a compact Lie group $G$, recovering prisms for $G=\mathbb Q_p/\mathbb Z_p$, presumably related to Dress-Siebeneicher's $G$-Witt vectors, such that $\pi^G_0$ of a nice $G$-$E_\infty$-ring spectrum is a $G$-prism.