A reduction theorem for Dade's projective conjecture
In this paper, the author presents a strategy to reduce Dade's conjectures to statements on quasisimple groups, by formulating the Character Triple Conjecture, which a strengthening of Dade's extended projective conjecture. Let \(p\) be a prime, \(d\) an integer, \(G\) a finite group with \(\mathrm{O}_p(G) \le Z(G)\), and \(B\) a \(p\)-block of \(G\) with a noncentral defect group. Denote by \(\mathfrak{P}(G|\mathrm{O}_p(G))\) the set of \(p\)-chains \(\mathbb{D}=(D_0= \mathrm{O}_p(G) \lneq D_1 \lneq \dots \lneq D_l)\) of \(G\) starting with \(\mathrm{O}_p(G)\). Here, \(l=|\mathbb{D}|\) is the length of \(\mathbb{D}\). The set of \(p\)-chains of even length is \(\mathfrak{P}(G|\mathrm{O}_p(G))_+\), and the set of \(p\)-chains of odd length is \(\mathfrak{P}(G|\mathrm{O}_p(G))_-\). Let \(G_\mathbb{D}\) be the normalizer of \(\mathbb{D}\) in \(G\), and \(B_\mathbb{D}\) the set of all \(p\)-blocks \(b\) of \(G_\mathbb{D}\) such that \(b^G = B\). For \(\chi \in \mathrm{Irr}(G)\), the defect \(\mathrm{d}(\chi)\) of \(\chi\) is defined by \(p^{\mathrm{d}(\chi)} \chi(1)_p = |G|_p\). For a set \(C\) of \(p\)-blocks of \(G\), the set \(\mathrm{Irr}^d(C)\) consists of all characters with defect \(d\) belonging to a block in \(C\). Denote by \(\mathcal{C}^d(B)_+\) the set of all pairs \((\mathbb{D},\theta)\) with \(\mathbb{D}\in \mathfrak{P}(G|\mathrm{O}_p(G))_+\) and \(\theta \in \mathrm{Irr}^d(B_\mathbb{D})\), and \(\mathcal{C}^d(B)_-\) is defined analogously. The group \(G\) acts on \(\mathcal{C}^d(B)_+\) and \(\mathcal{C}^d(B)_-\) by conjugation. For \((\mathbb{D},\theta)\in \mathcal{C}^d(B)_+\) denote by \(\overline{(\mathbb{D},\theta)}\) its \(G\)-orbit, and by \(\overline{\mathcal{C}^d(B)_+}\) and \(\overline{\mathcal{C}^d(B)_-}\) the corresponding sets of \(G\)-orbits. Dade's projective conjecture can be formulated as follows: There exists a bijection \(\Omega: \overline{\mathcal{C}^d(B)_+} \to \overline{\mathcal{C}^d(B)_-}\) such that \(\theta_{\mathrm{O}_p(G)}\) and \(\theta'_{\mathrm{O}_p(G)}\) are multiples of the same irreducible character whenever \((\mathbb{D},\theta)\in \mathcal{C}^d(B)_+\) and \((\mathbb{D}',\theta')\in \Omega(\overline{(\mathbb{D},\theta)})\). Dade has shown that the above conjecture for all groups implies the Alperin-McKay conjecture for all groups, and that it also implies the Alperin weight conjecture for all groups. Here, this conjecture strengthened by requiring that characters corresponding to each other via the above bijection give character triples that satisfy the equivalence relation ``\(\sim\)'' introduced in the paper. The relation roughly means that the two characters ``have the same Clifford theory'', in the context of blocks. The conjecture below adapts to Dade's conjecture the inductive conditions formulated for other local/global conjectures by Isaacs-Malle-Navarro, Navarro-Tiep and by the author. The character triple conjecture. Suppose that \(G \vartriangleleft A\). Then there exists an \(A_B\)-equivariant bijection \(\Omega: \overline{\mathcal{C}^d(B)_+} \to \overline{\mathcal{C}^d(B)_-}\) such that for every \((\mathbb{D},\theta)\in \mathcal{C}^d(B)_+\), some \((\mathbb{D},\theta')\in \Omega( \overline{(\mathbb{D},\theta)})\) satisfies \((A_{\mathbb{D},\theta}, G_\mathbb{D},\theta) \sim_G (A_{\mathbb{D}',\theta'}, G_{\mathbb{D}'} , \theta')\). It is shown here that this conjecture for a given group \(G\) implies Dade's extended projective conjecture for \(G\). It is also known that it holds for \(p\)-solvable groups. The main result of the paper is the following Theorem: Let \(\mathcal{S}\) be a set of simple nonabelian groups such that every covering group \(X\) of some \(S\in \mathcal{S}\) satisfies the character triple conjecture with respect to \(X \vartriangleleft X \rtimes \Aut(X)\). Then Dade's projective conjecture holds for every finite group \(G\) if every nonabelian simple group involved in \(G\) is contained in \(\mathcal{S}\). It is also shown that for simple groups with cyclic outer automorphism group, the above inductive condition is equivalent to Dade's invariant projective conjecture. This result also gives another approach to verifying the Alperin weight and the Alperin-McKay conjectures by using the classification of finite simple groups. The proof relies on the study and properties of the equivalence relation ``\(\sim\)'' on character triples. Then, the main steps of the proof are as follows: (i) A minimal counterexample \(G\) has a normal perfect subgroup \(K\) such that \(K/\mathrm{Z}(K)\) is the direct product \(S^r\) of groups isomorphic to a nonabelian simple group \(S\). This group and the centre of \(G\) generate the generalized Fitting subgroup. (ii) It is enough to consider the cardinality of character sets associated with the chains of \(K\). (iii) The universal covering of \(K\) is a group of the form \(\hat S^r\), where \(\hat S\) is the universal covering group of \(S\). The \(p\)-chains of \(K\) and the cardinalities of the character sets from (ii) can be studied via characters of subgroups of \(\hat S^r\rtimes \Aut(\hat S^r)\). (iv) Study the chains of \(\hat S\) and associated characters in the context of wreath product structure of \(\hat S^r\rtimes \Aut(\hat S^r)\). Examples are given where the character triple conjecture holds for covering groups of nonabelian simple groups. It also holds for most sporadic groups, for blocks with cyclic defect groups, and for the quasisimple groups \(\mathrm{SL}_2(q)\).
scientific article; zbMATH DE number 6705848
scientific article