# Slices for certain $C_p$-spectrum

By the work of Hill-Yarnall, for the group $$G=C_p,$$ all the slices for any spectrum, in particular, for $$S^V \wedge H\underline{\mathbb{Z}}$$, are classified. Here $$V$$ is a representation of $$C_p.$$

Again, following Yarnall's work, we know the spectrum $$S^n \wedge H\underline{\mathbb{Z}}$$ has the $$n$$-slice of the form $$S^{W(n)}\wedge H\underline{\mathbb{Z}}$$. Here $$W(n)$$ is a certain representation defined in Definition 3.5 of Yarnall's work "The slices of $$S^n \wedge H\underline{\mathbb{Z}}$$ for cyclic $$p$$-groups.

$$\mathbf{Question:}$$ If we take any $$C_p$$-representation $$V=m+n\xi$$. Is it true the $$\dim(V)$$-slice of the spectrum $$S^V \wedge H\underline{\mathbb{Z}}$$ is of the form $$S^{U(m,n)}\wedge H\underline{\mathbb{Z}}$$? If so, can we write explicitly what this representation $$U(m,n)$$ is (may be in terms of $$W(m)$$ and $$n$$)?

Thank you so much in advance. Any help will be appreciated.

This follows from the Hill-Yarnall formula for slices (I guess they do regular slices, so you have to deal with a shift if you want the classical ones). The reason is as follows: the slice of $$X$$ in dimension $$n$$ is given by first applying some algebraic procedure to $$\pi_WX$$ where $$W$$ is a certain representation of dimension $$n$$, and then suspending that Mackey functor by $$W$$. It turns out that, in our case, this algebraic procedure will always split out a Mackey functor which is equivalent to a suspension of $$\underline{\mathbb{Z}}$$.
Specifically, we have $$S^V \wedge \underline{\mathbb{Z}}$$ and you want the $$\mathrm{dim}(V)$$-slice. So we'll need to compute some $$\pi_{W-V}\underline{\mathbb{Z}}$$ where $$W-V$$ has dimension zero. You either get $$\underline{\mathbb{Z}}$$ or you get the transferred version $$\underline{\mathbb{Z}}_{\mathrm{tr}}$$ (when $$p=2$$ there is one further possibility, which is the Mackey functor that has $$\mathbb{Z}$$ on underlying with the sign representation and $$0$$ on fixed points). Now you do one of three possible algebraic procedures: (i) nothing, (ii) mod out the kernel of the restriction, or (iii) take the submackey functor generated by the transfer. The possible results of these algebraic procedures are again either $$\underline{\mathbb{Z}}$$ or $$\underline{\mathbb{Z}}_{\mathrm{tr}}$$ (or that extra possibility at $$p=2$$). So the $$\mathrm{dim}(V)$$-slice is given by the $$W$$-suspension of either $$\underline{\mathbb{Z}}$$ or $$\underline{\mathbb{Z}}_{\mathrm{tr}}$$. But now $$\underline{\mathbb{Z}}_{\mathrm{tr}} \simeq \Sigma^{2-\lambda}\underline{\mathbb{Z}}$$, so that's secretly also a suspension of $$\underline{\mathbb{Z}}$$. (Similarly, when $$p=2$$ you have the additional equivalence that $$\Sigma^{1-\sigma}\underline{\mathbb{Z}}$$ is the Mackey functor with $$0$$ fixed point part and $$\mathbb{Z}$$ with the sign representation on underlying).
You can work out what $$W$$ is explicitly (there's just gonna be several cases depending on writing the dimension of $$V$$ in the form $$rp+2k+\varepsilon$$ and splitting up into subcases depending on whether some quantity is positive or negative and so on...)