Möbius function of the subgroup lattice of a finite group and Euler characteristic (Q6604499)

Let \(V = \mathbb{F}_q^{n}\) be the finite vector space of dimension \(n\) over the finite field \(\mathbb{F}_q\) with \(q\) elements. Let \(G\) be a subgroup of \(\mathrm{GL}(V)\) acting irreducibly on \(V\). Let \(H\) be a subgroup of \(G\).\N\NThe main result (Theorem 4.5) of the paper is the following formula:\N\[\N- \mu_{\hat{\mathcal{I}}(G,H)}(H,G) = \sum_{E \in \Psi'(G,H)} (-1)^{|E|} = \sum_{X \in \Psi(G,H)} (-1)^{|X|} = - \tilde{\chi}(\Delta_1) = - \tilde{\chi}(\Delta_2).\N\]\NWe will explain the notation. For a finite poset \(\mathcal{P}\), let \(\mu_{\mathcal{P}}\) be the associated Möbius function. Let \(\Delta\) be a simplicial complex and let \(\chi(\Delta)\) be the Euler characteristic of \(\Delta\). The reduced Euler characteristic \(\tilde{\chi}(\Delta)\) of \(\Delta\) is defined by \(\tilde{\chi}(\emptyset) = 0\) and \(\tilde{\chi}(\Delta) = \chi(\Delta) -1\) if \(\Delta \not= \emptyset\).\N\NLet \N\[\N\mathcal{I}(G,H) = \{ K \leq G \mid H \leq K \leq M \ \mathrm{for \ some} \ M \in \mathcal{C}(G,H) \}\N\]\Nwhere \N\[\N\mathcal{C}(G,H) = \{ \mathrm{stab}_{G}(W) \mid 0 < W < V, \ H \leq \mathrm{stab}_{G}(W) \}.\N\]\NIf \(H\) acts reducibly on \(V\), then set \(\hat{\mathcal{I}}(G,H) = \mathcal{I}(G,H) \cup \{ G \}\), otherwise set \(\hat{\mathcal{I}}(G,H) = \{ H, G \}\). Set \(\Psi(G,H) = \{ X \subseteq \mathcal{C}(G,H) \mid \cap_{M \in X} M \not= H \}\) and \N\[\N\Psi'(G,H) = \{ E \subseteq S(V,H)^{*} \mid \bigcap_{W \in E} \mathrm{stab}_{G}(W) \not= H \}\N\]\Nwhere \(S(V,H)^{*} = S(V,H) \setminus \{ 0, V \}\) where \(S(V,H)\) is the lattice of \(H\)-invariant subspaces of \(V\).\N\NLet \(\Delta_1\) be the simplicial complex with vertex set given by the subspaces \(W \in S(V,H)^{*}\) for which \(H \not= \mathrm{stab}_{G}(W)\) with set of faces given by \(\Psi'(G,H)\). Finally, let \(\Delta_2\) be the simplicial complex with vertex set given by the subgroups \(M \in \mathcal{C}(G,H)\) such that \(H \not= M\) with set of faces given by \(\Psi(G,H)\).\N\NA motivation for the paper was the PhD thesis of \textit{J. W. Shareshian} [Combinatorial properties of subgroup lattices of finite groups. Ann Arbor, MI: Rutgers, The State University of New Jersey (PhD Thesis) (1996)] where the problem of computing \(\mu(1,G) = \mu_{\mathcal{P}}(1,G)\) where \(\mathcal{P}\) is the subgroup lattice of a finite classical group \(G\) is considered. The work is also related to a well-known conjecture of \textit{A. Mann} [Int. J. Algebra Comput. 15, No. 5--6, 1053--1059 (2005; Zbl 1098.20025)]. Let \(G\) be a PFG-group. Let \(\mu\) be the Möbius function on the lattice of open subgroups of \(G\). The number \(|\mu(H,G)|\) is bounded by a polynomial function in the index \(|G:H|\) of \(H\) in \(G\) and the number of subgroups \(H\) of \(G\) of index \(m\) with \(\mu(H,G) \not= 0\) grows at most polynomially in \(m\). In [J. Group Theory 27, No. 2, 275--296 (2024; Zbl 07812017)], the second author uses Theorem 4.5 to handle a special case of this conjecture.