A stronger version of a problem of Kenneth Brown using representations Let $G$ be a finite group and $\mathcal{L}(G)$ its subgroup lattice. Let $\mu$ be the Möbius function on $\mathcal{L}(G)$.
The reduced Euler characteristic of the order complex of the coset poset $\{ Kg \ | \ K<G, \ g \in G \} $ is $$\chi(G) := -\sum_{H \in \mathcal{L}(G)} \mu(H,G)|G:H|.$$ Gaschütz showed that $\chi(G)$ is nonzero if $G$ is solvable and the question whether  $\chi(G)$ is nonzero for every finite group $G$ is an open problem motivated by Brown (see DOI: 10.1016/j.aim.2015.10.018).
There is a relative generalization of this problem. Let $H$ be a subgroup of $G$. The reduced Euler characteristic of the order complex of the coset poset $\{ Kg \ | \ K \in [H,G), \ g \in G \} $ is $$\chi(H,G) := -\sum_{K \in [H,G]} \mu(K,G)|G:K|.$$
The question whether  $\chi(H,G)$ is nonzero for every interval of finite groups $[H,G]$ is also open.  
Now for any representation $V$ of $G$, and any subgroup $K$, consider the fixed point subspace:
$$ V^K := \{v \in V \ | \ k\cdot v = v \ , \forall k \in K  \}  $$ 
Let $V_1, \dots , V_n$ be (equivalence class representatives of) the irreducible complex representations of $G$. By Frobenius reciprocity, we have the following identity (see here):
$$|G:K| = \sum_{i=1}^n \dim(V_i)\dim(V_i^K)$$
It follows that:  $$\chi(H,G) = \sum_{i=1}^n \dim(V_i) \chi_i(H,G).$$
with $$\chi_i(H,G) := -\sum_{K \in [H,G]} \mu(K,G)\dim(V_i^K)$$
So, a stronger version of the above open problem, for a given interval $[H,G]$, is the following:  
Question: Is there $i$ with $\chi_i(H,G)$ nonzero, and with all nonzero $\chi_j(H,G)$ of same sign?
It is checked (by GAP) for $|G| \le 215$ (except $128,192$), and for any simple group $G$ with $|G| \le 504$.
 A: Not an actual answer, but a comment doesn't offer much space.
I have recently added a couple of formulas involving sums of $\chi(H,G)$, with $H$ running over various subgroups, to my paper Elementary Proof of a Theorem of Hawkes, Isaacs and Özaydin. An example is $\Sigma_{g\in G}\,\chi(\langle g\rangle,G)=0$ for $G>1$. It turns out that $g\mapsto\chi(\langle g\rangle,G)$ is a virtual character of $G$, often an actual character or the negative of one, though mixed-sign cases occur as well.
The notation used for $\chi(H,G)$ in the paper is $-\varphi^{-1}(H,G)$, where $\varphi$ is an element, which is related to the Eulerian function of $G$, of the incidence algebra over $\mathbb{Z}$ of $\mathcal{L}(G)$, and the inverse is taken in that ring.
It appears to be the case that $\chi(\langle g\rangle,G)=\Sigma_{H\leq G,\,\langle H,g\rangle=G}\,\chi(1,H)\,[G:H]$ for all $g\in G$. I have verified this using GAP for some 250 groups of varying nature.
Edit: managed to prove this generally, see the latest edition of the paper.
