Search

This is a contribution to the study of $\mathrm {Irr}(G)$ as an $\mathrm {Aut}(G)$ -set for G a finite quasisimple group. Focusing on the last open case of groups of Lie type $\mathrm {D}$ and $^2\mathrm {D}$ , a crucial property is the so-called $A'(\infty)$ condition expressing that diagonal automorphisms and graph-field automorphisms of G have transversal orbits in $\mathrm {Irr}(G)$. This is part of the stronger $A(\infty)$ condition introduced in the context of the reduction of the McKay conjecture to a question about quasisimple groups. Our main theorem is that a minimal counterexample to condition $A(\infty)$ for groups of type $\mathrm {D}$ would still satisfy $A'(\infty)$. This will be used in a second paper to fully establish $A(\infty)$ for any type and rank. The present paper uses Harish-Chandra induction as a parametrization tool. We give a new, more effective proof of the theorem of Geck and Lusztig ensuring that cuspidal characters of any standard Levi subgroup of $G=\mathrm {D}_{ l,\mathrm {sc}}(q)$ extend to their stabilizers in the normalizer of that Levi subgroup. This allows us to control the action of automorphisms on these extensions. From there, Harish-Chandra theory leads naturally to a detailed study of associated relative Weyl groups and other extendibility problems in that context. [ABSTRACT FROM AUTHOR]

We show that the decomposition matrix of a given group G is unitriangular, whenever G has a normal subgroup N such that the decomposition matrix of N is unitriangular, G/N is abelian and certain characters of N extend to their stabilizer in G. Using the recent result by Brunat–Dudas–Taylor establishing that unipotent blocks have a unitriangular decomposition matrix, this allows us to prove that blocks of groups of quasi-simple groups of Lie type have a unitriangular decomposition matrix whenever they are related via Bonnafé–Dat–Rouquier's equivalence to a unipotent block. This is then applied to study the action of automorphisms on Brauer characters of finite quasi-simple groups. We use it to verify the so-called inductive Brauer–Glauberman condition that aims to establish a Glauberman correspondence for Brauer characters, given a coprime action. [ABSTRACT FROM AUTHOR]

We give a criterion that simplifies the checking of the inductive Alperin weight condition for the remaining open cases of simple groups of Lie type (see Theorem 3.3). It is strongly related in form to the criterion of the second author for the inductive McKay conditions (see Späth, Bull. Lond. Math. Soc. 44 (2012), no. 3, 426–438, 2.12) that has proved very useful. The proof follows from a Clifford theory for weights intrinsically present in the proof of reduction theorems of the Alperin weight conjecture given by Navarro–Tiep and the second author. We also give a related criterion for the inductive blockwise Alperin weight condition (Theorem 4.5). [ABSTRACT FROM AUTHOR]

We introduce H-triples and a partial order relation on them, generalizing the theory of ordering character triples developed by Navarro and Späth. This generalization takes into account the action of Galois automorphisms on characters and, together with previous results of Ladisch and Turull, allows us to reduce the Galois-McKay conjecture to a question about simple groups. [ABSTRACT FROM AUTHOR]

Suppose that B is a Brauer p-block of a finite group G with a unique modular character φ .We prove that φ is liftable to an ordinary character of G (which moreover is p-rational for odd p). This confirms the basic set conjecture for these blocks. [ABSTRACT FROM AUTHOR]

We verify the inductive McKay condition for simple groups of Lie type C, namely finite projective symplectic groups. This contributes to the program of a complete proof of the McKay conjecture for all finite groups via the reduction theorem of Isaacs-Malle-Navarro and the classification of finite simple groups. In an important step we use a new counting argument to determine the stabilizers of irreducible characters of a finite symplectic group in its outer automorphism group. This is completed by analogous results on characters of normalizers of Sylow d-tori in those groups. [ABSTRACT FROM AUTHOR]

As a step to establish the McKay conjecture on character degrees of finite groups, we verify the inductive McKay condition introduced by Isaacs-Malle-Navarro for simple groups of Lie type An-1, split or twisted. Key to the proofs is the study of certain characters of SLn(q) and SUn(q) related to generalized Gelfand-Graev representations. As a by-product we can show that a Jordan decomposition for the characters of the latter groups is equivariant under outer automorphisms. Many ideas seem applicable to other Lie types. [ABSTRACT FROM AUTHOR]

In this paper, we propose a strengthening of Dade's Conjecture. This version, called the Character Triple Conjecture, once assumed for quasisimple groups, is shown to imply Dade's Projective Conjecture for all finite groups. In particular Dade's Projective Conjecture holds for a group whose nonabelian simple sections have only covering groups satisfying the Character Triple Conjecture. We verify the new conjecture for some classes of quasisimple groups. [ABSTRACT FROM AUTHOR]

We prove several results giving substantial evidence in support of the conjectural existence of a Glauberman-Isaacs bijection for Brauer characters under a coprime action. We also discuss related bijections for the McKay conjecture. [ABSTRACT FROM AUTHOR]

We verify the inductive Blockwise Alperin Weight (BAW) and the inductive Alperin-McKay (AM) conditions introduced by the second author, for p-blocks of finite quasisimple groups with cyclic defect groups whenever the prime p is odd. Furthermore, we establish a criterion that describes conditions under which the inductive AM condition for blocks with abelian defect groups implies the inductive BAW condition for those blocks. [ABSTRACT FROM AUTHOR]

In this paper, the unproven half of Richard Brauer's Height Zero Conjecture is reduced to a question on simple groups. [ABSTRACT FROM AUTHOR]

As a first step in a general verification of the Alperin-McKay conjecture, we prove a reduction of the conjecture to statements about simple groups. Furthermore we show in numerous cases, that simple groups satisfy the required conditions. The methods are also applied to obtain similar results for refinements due to Isaacs and Navarro. [ABSTRACT FROM AUTHOR]

We show that the blockwise version of the Alperin weight conjecture is true if for every finite non-abelian simple group a set of conditions holds. Furthermore we prove that several series of simple groups satisfy these assumptions. This refines recent work of Navarro and Tiep, who proved an analogous reduction theorem for the non-blockwise version of the Alperin weight conjecture. [ABSTRACT FROM AUTHOR]

We reformulate the inductive McKay condition, from Isaacs–Malle–Navarro, and apply the new criterion to simple groups of Lie type, when the prime is the defining characteristic p. Using a recent result of Maslowski, we obtain that these simple groups satisfy the inductive McKay condition for p. In the non-simply laced types and exceptional types, this reproves earlier results by Brunat and Brunat–Himstedt. [ABSTRACT FROM PUBLISHER]