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Now forgetting down to spectra with a $C_p$-action (i.e., the homotopy theory of $C_p$-diagrams in spectra or, alternatively, the $\infty$-category $\mathrm{Fun}(BC_p, \mathrm{Sp}))$ and taking rational homology we obtain an isomorphism of two graded $\mathbb{Q}$-vector spaces with $C_p$-actions. This shows that the permutation $C_p$-representation $\mathbb{Q}[C_p]$ is in the localizing subcategory generated by the trivial representation (since $p$ is odd, the homology of the representation spheres is always a shifted copy of $\mathbb{Q}$ with a trivial action). We are unable to build a non-trivial action from trivial ones, so this is a contradiction.

I do not have a good answer for the remaining questions. For general groups there are many indices in the $RO(G)$-grading. Even if you pass to the smaller $JO(G)$-grading (which is what you get when you quotient $RO(G)$ by the equivalence relation $V\sim W\iff S^V\simeq S^W$), I would think it would be difficult to convey the data concisely. For cyclic $p$-groups (and perhaps more), one can probably get at the $RO(G)$-graded groups for an Eilenberg-MacLane functor. Ferland-Lewis and Lewis have relevant calculations. I imagine there are more calculations when $G=C_2$.

Now forgetting down to spectra with a $C_p$-action (i.e., the homotopy theory of $C_p$-diagrams in spectra or, alternatively, the $\infty$-category $\mathrm{Fun}(BC_p, \mathrm{Sp}))$ and taking rational homology we obtain an isomorphism of two graded $\mathbb{Q}$-vector spaces with $C_p$-actions. This shows that the permutation $C_p$-representation $\mathbb{Q}[C_p]$ is in the localizing subcategory generated by the trivial representation (since $p$ is odd, the homology of the representation spheres is always a shifted copy of $\mathbb{Q}$ with a trivial action). Now we have a cofiber sequence $$\mathbb{Q}[C_p]^{C_p}\to \mathbb{Q}[C_p]\to \mathbb{Q}[C_p]/(\mathbb{Q}[C_p]^{C_p}).$$ Now $\mathbb{Q}[C_p]$ is in the localizing subcategory generated by trivial representations if and only if the cofiber is. However the only map from a trivial rep to the non-trivial cofiber is 0 (note this would fail in characteristic $p$), so the cofiber fails to be in the localizing subcategory generated by the trivial reps and we obtain a contradiction.

I do not have a good answer for the remaining questions. For general groups there are many indices in the $RO(G)$-grading. Even if you pass to the smaller $JO(G)$-grading (which is what you get when you quotient $RO(G)$ by the equivalence relation $V\sim W\iff S^V\simeq S^W$), I would think it would be difficult to convey the data concisely. For cyclic $p$-groups (and perhaps more), one can probably get at the $RO(G)$-graded groups for an Eilenberg-MacLane spectrum associated to a Mackey functor. Ferland-Lewis and Lewis have relevant calculations. I imagine there are more calculations when $G=C_2$.

I do not have a good answer for the remaining questions. For general groups there are many indices in the $RO(G)$-grading. Even if you pass to the smaller $JO(G)$-grading, I would think it would be difficult to convey the data concisely. For cyclic $p$-groups (and perhaps more), one can probably get at the $RO(G)$-graded groups for an Eilenberg-MacLane functor. Ferland-Lewis and Lewis have relevant calculations. I imagine there are more calculations when $G=C_2$.

I do not have a good answer for the remaining questions. For general groups there are many indices in the $RO(G)$-grading. Even if you pass to the smaller $JO(G)$-grading (which is what you get when you quotient $RO(G)$ by the equivalence relation $V\sim W\iff S^V\simeq S^W$), I would think it would be difficult to convey the data concisely. For cyclic $p$-groups (and perhaps more), one can probably get at the $RO(G)$-graded groups for an Eilenberg-MacLane functor. Ferland-Lewis and Lewis have relevant calculations. I imagine there are more calculations when $G=C_2$.

Indeed by definition, a map $f\colon X\to Y $ is a $G$-equivalence, if it induces an isomorphism $$f_*\colon [\Sigma^* G/H_+, X]\to [\Sigma^* G/H_+, Y],$$ for every (closed) subgroup $H\subset G$. When $G=C_p$ the only subgroups are the trivial subgroup and the whole group $G$. Under the hypothesis of 1. the map $f_*$ is an isomorphism by assumption when $H=G$, so we just need to show $G/e_+={C_p}_+\in \mathcal{C}_f$. This directly relations to condition 2.: The smallest localizing category containing $S^0\in \mathrm{Lin}$ and ${C_p}_+$ is $\mathrm{Sp}_{C_p}$.

Suppose that $p$ is odd and 2. holds. Let $g\colon Z\to {C_p}_+$ be a $\mathrm{Lin}$-cellularization of ${C_p}_+$. In other words, it is a colocalization with respect to $\mathrm{Lin}$; $Z$ is constructed by iteratively gluing 'cells' from $\mathrm{Lin}$ (just as in a CW-approximation) and $g$ induces an isomorphism on $RO(G)$-graded homotopy groups. Since we are assuming 2. $g$ is actually a $G$-equivalence.

Now forgetting down to spectra with a $C_p$-action (i.e., the homotopy theory of $C_p$-diagrams in spectra or the $\infty$-category $\mathrm{Fun}(BC_p, \mathrm{Sp}))$ and taking rational homology we obtain an isomorphism of two graded $\mathbb{Q}$-vector spaces with $C_p$-actions. This shows that the permutation $C_p$-representation $\mathbb{Q}[C_p]$ is in the localizing subcategory generated by the trivial representation (since $p$ is odd, the homology of the representation spheres is always a shifted copy of $\mathbb{Q}$ with a trivial action). We are unable to build a non-trivial action from trivial ones, so this is a contradiction.

If instead of working over the sphere, we work in the category of $KU_G$-modules, then I can say a bit more. Let $G$ be a connected compact Lie group with $\pi_1 G$ torsion-free. Then it is a result of Akhil Mathew, Niko Naumann, and myself that a map $f\colon X\rightarrow Y$ of $KU_G$-modules is an equivalence precisely when it induces isomorphisms: $$ f_*\colon [\Sigma^* S^0, X]\to [\Sigma^* S^0, Y].$$ So here, you don't even need all of the representation spheres, just the trivial representations. This result follows from Thm. 8.3 in Nilpotence and descent in equivariant stable homotopy theory.

Indeed by definition, a map $f\colon X\to Y $ is a $G$-equivalence, if it induces an isomorphism $$f_*\colon [\Sigma^* G/H_+, X]\to [\Sigma^* G/H_+, Y],$$ for every (closed) subgroup $H\subset G$. When $G=C_p$, the only subgroups are the trivial subgroup and the whole group $G$. Under the hypothesis of 1., the map $f_*$ is an isomorphism by assumption when $H=G$, so we just need to show $G/e_+={C_p}_+\in \mathcal{C}_f$. This directly relates to condition 2.: The smallest localizing category containing $S^0\in \mathrm{Lin}$ and ${C_p}_+$ is $\mathrm{Sp}_{C_p}$.

Suppose that $p$ is odd and 2. holds, we will derive a contradiction. Let $g\colon Z\to {C_p}_+$ be a $\mathrm{Lin}$-cellularization of ${C_p}_+$. In other words, it is a colocalization with respect to $\mathrm{Lin}$; $Z$ is constructed by iteratively gluing 'cells' from $\mathrm{Lin}$ (just as in a CW-approximation) and $g$ induces an isomorphism on $RO(G)$-graded homotopy groups. Since we are assuming 2., $g$ is actually a $G$-equivalence.

Now forgetting down to spectra with a $C_p$-action (i.e., the homotopy theory of $C_p$-diagrams in spectra or, alternatively, the $\infty$-category $\mathrm{Fun}(BC_p, \mathrm{Sp}))$ and taking rational homology we obtain an isomorphism of two graded $\mathbb{Q}$-vector spaces with $C_p$-actions. This shows that the permutation $C_p$-representation $\mathbb{Q}[C_p]$ is in the localizing subcategory generated by the trivial representation (since $p$ is odd, the homology of the representation spheres is always a shifted copy of $\mathbb{Q}$ with a trivial action). We are unable to build a non-trivial action from trivial ones, so this is a contradiction.

If instead of working over the sphere, we work in the category of $KU_G$-modules, then I can say a bit more. Let $G$ be a connected compact Lie group with $\pi_1 G$ torsion-free. Then it is a result of Akhil Mathew, Niko Naumann, and myself that a map $f\colon X\rightarrow Y$ of $KU_G$-modules is an equivalence precisely when it induces isomorphisms: $$ f_*\colon [\Sigma^* S^0, X]\to [\Sigma^* S^0, Y].$$ So here, you don't even need all of the representation spheres, just the trivial representations. This result follows from Thm. 8.3 of Nilpotence and descent in equivariant stable homotopy theory.