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I'll first explain what Möbius 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 Möbius 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 $v(p) = \sum_{q \geq p} u(q)$. Möbius 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 simplicial 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?

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  • $\begingroup$ Being rather ignorant of representation theory as it is formally taught, I am sure this formulation is new to me. However, almost any article talking about this would refer to earlier work (Rota, maybe?) on such inversion over posets. Maybe a citation index and an email to order theorists (perhaps J.D. Farley) would help? I would be surprised at such an article without such citation, and I can imagine this article as part of a conference collection. Gerhard "Assuming No Good Answer Here" Paseman, 2018.07.22. $\endgroup$ Jul 22, 2018 at 15:11
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    $\begingroup$ @TomLeinster might know about this - he has done work on Möbius inversion on categories, and it might be that if you make a category out of $G$ and $P$ (with morphisms $(g,p,q)$ for $gp\leqslant q$ or something like that) then what you ask might be an instance of what he does... $\endgroup$ Jul 22, 2018 at 17:56
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    $\begingroup$ I guess that you want $(U(p))_{p\in P}$ to be a $G$-equivariant representation of the poset $P$. That is for all $p\in P$, one is given a finite dimensional vector space $U(p)$ together with linear maps $r__p^q$ : $U(q)\rightarrow U(p)$ whenever $q\geqslant p$, and maps $f_{g,p}$~: $U(p)\rightarrow U(g.p)$, these data being subject to some obvious compatibility conditions. $\endgroup$ Jul 23, 2018 at 6:38
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    $\begingroup$ This is somewhat related to the content of section 2.2 in arxiv.org/pdf/math/0411461.pdf (joint paper of myself and Uri Onn). $\endgroup$
    – Uri Bader
    Jul 23, 2018 at 6:47
  • $\begingroup$ I would perhaps look for a reference to a suitably homological/topological formulation of the Möbius inversion theorem, where equivariance becomes obvious from the constructions in the proof being functorial. In particular one can formulate Möbius inversion in terms of sheaf cohomology on posets, where we think of the poset as a topological space with the Alexandroff topology and a functor from $P$ to vector spaces as defining a sheaf on this space. Then the terms in the Möbius inversion formula can be given topological meaning and there is a spectral sequence proof of the formula. $\endgroup$ Jul 23, 2018 at 19:22

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Sami Assaf and I prove this in section 5 of our paper Specht modules decompose as alternating sums of restrictions of Schur modules. It is surprising that we couldn't find a reference!

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This is more of a long comment. I am not sure I understand your construction, but the sort of alternating sum you take has a preimage in the Burnside ring, and is often called the "Lefschetz invariant" by finite group theorists. One of the Representation and Cohomology books by Benson has a section about this. Also see this paper by Thévenaz.

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