I'll first explain what Mobius inversion says, and then state what I am fairly sure the equivariant version is. I can write out a proof, but I also can't believe this hasn't been done already; this is a request for references to where it has already been done.


**Ordinary Mobius Inversion** Let $P$ be a finite poset with minimal element $0$. Let $u$ be a function $P \to \mathbb{Z}$ and define $v: P \to \mathbb{Z}$ by $v(p) = \sum\_{q \geq p} u(q)$. Mobius inversion aims to recover $u(0)$ from the values of $v$. It says that $u(0) = \sum\_{q \in P} \mu(q) v(q)$. The function $\mu : P \to \mathbb{Z}$ can be described topologically: Let $(0,q)$ be the poset $\{ r \in P : 0 < r < q \}$ and let $\Delta((0,q))$ be the order complex, which is the simiplicial complex whose faces are totally ordered subsets of $(0,q)$. Then $\mu(q)$ is the reduced ordered characteristic of $\Delta((0,q))$. 


**The equivariant situation** Let $P$ be a finite poset with minimal element $0$ and let $G$ be a group acting on $P$. For each $p \in P$, let $U(p)$ be a finite dimensional $\mathbb{C}$-vector space. Define $V(p) : = \bigoplus\_{q \geq p} U(p)$, so $V(0) = \bigoplus\_p U(p)$. Let $G$ act on $V(0)$, with $g U(p)= U(gp)$. My goal is to recover the class of $U(0)$, in the representation ring $Rep(G)$, from the $V(p)$'s. 


For $p \in P$, let $G\_p$ be the stabilizer of $p$, so $U(p)$ and $V(p)$ are $G\_p$-reps. Let $\mu\_{eq}(q)$ be the equivariant reduced Euler characteristic of $q$, meaning the sum $\sum (-1)^j [\tilde{H}^j(\Delta((0,q)))]$ computed in the representation ring $Rep(G\_q)$. Let $G\backslash P$ be a set of orbit representatives for $G$ acting on $P$. 


Then I claim that
$$[U(0)] = \sum\_{q \in G \backslash P} \mathrm{Ind}\_{G\_q}^G \left[ \mu\_{eq}(q) \otimes V(p) \right]$$
in $Rep(G)$.


Has anyone seen this before?



Equivariant Mobius inversion
Sami Assaf and I prove this in section 5 of our paper [Specht modules decompose as alternating sums of restrictions of Schur modules](https://arxiv.org/abs/1809.10125). It is surprising that we couldn't find a reference!


