# 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 any 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 any 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 $|G| \le 504$.