Let $G$ be an arbitrary finite group and fix a prime number $p$. The McKay conjecture asserts that $G$ and the normalizer in $G$ of a Sylow $p$-subgroup have equal numbers of irreducible characters with degrees not divisible by $p$. The Alperin-McKay conjecture is a version of this as applied to individual Brauer $p$-blocks of $G$. We offer evidence that perhaps much stronger forms of both of these conjectures are true.
The so-called "local-global" conjectures in the representation theory of finite groups relate the representation theory of $G$ to that of certain proper subgroups, such as the normalizers of particular $p$-groups. Recent results by several authors reduce some of these conjectures to showing that a certain collection of stronger conditions holds for all finite simple groups. Here, we show that $G=Sp_6(2^a)$ is "good" for these reductions for the McKay conjecture, the Alperin weight conjecture, and their blockwise versions.
The mathematical software \texttt{GAP} (Groups, Algorithms, Programming) offers a powerful set of tools to investigate computationally group theory. Using this software package we investigate a variation of a well-known problem in representation theory, the McKay Conjecture, that asserts that there is a bijection between two sets of complex-valued functions defined on two finite groups. In order to do so, we use the extensive group database from \texttt{GAP} and Kuhn's Algorithm in order to test our conjecture.
We refine the reduction theorem of the McKay Conjecture proved by Isaacs, Malle and Navarro. Assuming the inductive McKay condition, we obtain a strong version of the McKay Conjecture that gives central isomorphic character triples.
In this paper we consider the inductive Alperin-McKay condition for quasi-isolated 2-blocks of exceptional groups of Lie type. Thereby, we complete the proof of the Alperin-McKay conjecture for the prime 2.
We show that Dade's ordinary conjecture implies the Alperin-McKay conjecture. We remark that some of the methods can be used to identify a canonical height zero character in a nilpotent block.
In this paper, we show that the Alperin-McKay conjecture holds for 2-blocks of maximal defect. A major step in the proof is the verification of the inductive Alperin-McKay condition for the principal 2-block of groups of Lie type in odd characteristic.
Let $G$ be an arbitrary finite group. The McKay conjecture asserts that $G$ and the normaliser $N_G (P)$ of a Sylow $p$-subgroup $P$ in $G$ have the same number of characters of degree not divisible by $p$ (that is, of $p'$-degree). We propose a new refinement of the McKay conjecture, which suggests that one may choose a correspondence between the characters of $p'$-degree of $G$ and $N_G (P)$ to be compatible with induction and restriction in a certain sense. This refinement implies, in particular, a conjecture of Isaacs and Navarro. We also state a corresponding refinement of the Broué abeli…
The Alperin-McKay conjecture is a longstanding open conjecture in the representation theory of finite groups. Späth showed that the Alperin-McKay conjecture holds if the so-called inductive Alperin-McKay (iAM) condition holds for all finite simple groups. In a previous paper, the author has proved that it is enough to verify the inductive condition for quasi-isolated blocks of groups of Lie type. In this paper we show that the verification of the iAM-condition can be further reduced in many cases to isolated blocks. As a consequence of this we obtain a proof of the Alperin-McKay conjecture for…
We generalize the theory of ordering character triples, developed by Navarro and Späth, by taking into account the action of Galois automorphisms on characters. This new technique, together with previous results of Ladisch and Turull, allows us to reduce the Galois--McKay conjecture to a question about simple groups.
We prove for finite reductive groups $G$ of classical type, that every irreducible character of $L$ extends to its inertia group in $N$, where $L$ is an abelian centraliser of a Sylow $d$-torus $\mathbf S$ of $G$ and $N:=N_G(\mathbf S)$. This gives a precise description of the irreducible characters of $N$. Furthermore it enables us to verify the McKay conjecture in this situation for $G$ and some primes.
In this paper characters of the normaliser of $d$-split Levi subgroups in $\mathrm {SL}_n(q)$ and $\mathrm {SU}_n(q)$ are parametrized with a particular focus on the Clifford theory between the Levi subgroup and its normalizer.These results are applied to verify the Alperin-McKay conjecture for primes $\ell$ with $\ell\nmid 6(q^2-1)$ and the Alperin weight conjecture for $\ell$-blocks of those quasi-simple groups with abelian defect. The inductive Alperin-McKay condition and inductive Alperin weight condition by the second author are verified for certain blocks of $\mathrm {SL}_n(q)$ and $\mat…
For a prime $\ell$, the McKay conjecture suggests a bijection between the set of irreducible characters of a finite group with $\ell'$-degree and the corresponding set for the normalizer of a Sylow $\ell$- subgroup. Navarro's refinement suggests that the values of the characters on either side of this bijection should also be related, proposing that the bijection commutes with certain Galois automorphisms. Recently, Navarro--Spaeth--Vallejo have reduced the McKay--Navarro conjecture to certain "inductive" conditions on finite simple groups. We prove that these inductive McKay--Navarro (also c…
The Alperin-McKay conjecture relates height zero characters of an $\ell$-block with the ones of its Brauer correspondent. This conjecture has been reduced to the so-called inductive Alperin-McKay conditions about quasi-simple groups by the third author. Those conditions are still open for groups of Lie type. The present paper describes characters of height zero in $\ell$-blocks of groups of Lie type over a field with $q$ elements when $\ell$ divides $q-1$. We also give information about $\ell$-blocks and Brauer correspondents. As an application we show that quasi-simple groups of type $C$ over…
We prove that for most groups of Lie type, the bijections used by Malle and Spaeth in the proof of Isaacs-Malle-Navarro's inductive McKay conditions for the prime 2 and odd primes dividing q - 1 are also equivariant with respect to certain Galois automorphisms. In particular, this shows that these bijections are candidates for proving Navarro-Spaeth-Vallejo's recently-posited inductive Galois-McKay conditions. On the way, we show that several simple groups of Lie type satisfy the McKay--Navarro conjecture for the prime 2.
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 $A_{n-1}$, split or twisted. Key to the proofs is the study of certain characters of SL$_n(q)$ and SU$_n(q)$ 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.
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.
We prove the McKay conjecture on characters of odd degree. A major step in the proof is the verification of the inductive McKay condition for groups of Lie type and primes $\ell$ such that a Sylow $\ell$-subgroup or its maximal normal abelian subgroup is contained in a maximally split torus by means of a new equivariant version of Harish-Chandra induction. Specifics of characters of odd degree, namely that they only lie in very particular Harish-Chandra series then allow us to deduce from it the McKay conjecture for the prime~$2$, hence for characters of odd degree.
Let $G$ be a finite group, and $α$ a nontrivial character of $G$. The McKay graph $\mathcal{M}(G,α)$ has the irreducible characters of $G$ as vertices, with an edge from $χ_1$ to $χ_2$ if $χ_2$ is a constituent of $αχ_1$. We study the diameters of McKay graphs for finite simple groups $G$. For alternating groups, we prove a conjecture made in [LST]: there is an absolute constant $C$ such that $\hbox{diam}\,{\mathcal M}(G,α) \le C\frac{\log |\mathsf{A}_n|}{\log α(1)}$ for all nontrivial irreducible characters $α$ of $\mathsf{A}_n$. Also for classsical groups of symplectic or orthogonal type of …
Let $p$ be an odd prime number. In this paper, we characterize the nonabelian composition factors of a finite group with odd $p$-Sylow automizers, and then prove that the McKay conjecture, the Alperin weight conjecture and the Alperin-McKay conjecture hold for such group.
Let p be a prime, B a p-block of a finite group G and b its Brauer correspondent. According to the Alperin-McKay Conjecture, there exists a bijection between the set of irreducible ordinary characters of height zero of B and those of b. In this paper, we show that whenever G is p-solvable such a bijection can be found, both for ordinary and Brauer characters, with the additional property of being compatible with divisibility of character degrees. In this case, we also show that the dimension of b divides the dimension of B.
We show that Brauer's height zero conjecture holds for blocks of finite quasi-simple groups. This result is used in Navarro-Späth's reduction of this conjecture for general groups to the inductive Alperin-McKay condition for simple groups.
Recently Navarro proposed a strengthening of the unsolved McKay conjecture using Galois automorphisms. We prove that the Navarro conjecture holds for the alternating groups when the prime p is odd.
We determine the non-abelian composition factors of the finite groups with Sylow normalizers of odd order. As a consequence, among others, we prove the McKay conjecture and the Alperin weight conjecture for these groups.
The so-called inductive McKay condition on finite simple groups, due to Isaacs-Malle-Navarro (2007), has been recently reformulated by Späth. We show that this reformulation applies to the reduction theorem for Alperin's weight conjecture, due to Navarro-Tiep (2011). This also simplifies the checking of the inductive condition for Alperin's weight conjecture in the case of simple groups of Lie type with regard to the defining prime.
In [7], G. Navarro proposed a refinement of the McKay conjecture involving a special class of Galois automorphisms. In [6] this new conjecture was verified by the author for the alternating groups A(n) when p=2. In this note the Navarro conjecture is verified for the p-singular characters of the principal block of the alternating groups in the case of abelian defect when p is odd.
We give a summary of R. Borcherds' solution (with some modifications) to the following part of the Conway-Norton conjectures: Given the Monster simple group and Frenkel-Lepowsky-Meurman's moonshine module for the group, prove the equality between the graded characters of the elements of the Monster group acting on the module (i.e., the McKay-Thompson series) and the modular functions provided by Conway and Norton. The equality is established using the homology of a certain subalgebra of the monster Lie algebra, and the Euler-Poincare identity.
We show that the refinement of Alperin's Conjecture proposed in "Frobenius Categories versus Brauer Blocks", Progress in Math. 274, can be proved by checking that this refinement holds on any central k*-extension of a finite group H containing a normal simple group S with trivial centralizer in H and p'-cyclic quotient H/S. This paper improves our result in [ibidem, Theorem 16.45] and repairs some bad arguments there.
Let $p$ be a prime and $G$ a finite group. A complex character of $G$ is called almost $p$-rational if its values belong to a cyclotomic field $\mathbb{Q}(e^{2πi/n})$ for some $n\in \mathbb{Z}^+$ prime to $p$ or precisely divisible by $p$. We prove that, in contrast to usual $p$-rational characters, there are always "many" almost $p$-rational irreducible characters in finite groups. We obtain both explicit and asymptotic bounds for the number of almost $p$-rational irreducible characters of $G$ in terms of $p$. In fact, motivated by the McKay-Navarro conjecture, we obtain the same bound for th…
This paper is a contribution to the general program introduced by Isaacs, Malle and Navarro to prove the McKay conjecture in the representation theory of finite groups. We develop new methods for dealing with simple groups of Lie type in the defining characteristic case. Using a general argument based on the representation theory of connected reductive groups with disconnected center, we show that the inductive McKay condition holds if the Schur multiplier of the simple group has order 2. As a consequence, the simple groups \Orth_{2m+1}(p^n) and PSp_{2m}(p^n) are "good" for p>2 and the simple …
Let $p$ be a prime and $G$ a finite group. We propose a strong bound for the number of $p'$-degree irreducible characters of $G$ in terms of the commutator factor group of a Sylow $p$-subgroup of $G$. The bound arises from a recent conjecture of Navarro and Tiep [NT21] on fields of character values and a phenomenon called the continuity of $p$-rationality level of $p'$-degree characters. This continuity property in turn is predicted by the celebrated McKay-Navarro conjecture [Nav04]. We achieve both the bound and the continuity property for $p=2$.
We characterize finite groups having a cyclic Sylow p-subgroup in terms of the action of a specific Galois automorphism on the principal p-block for p=2,3. We show that the analog statement for blocks with arbitrary defect group would follow from the blockwise McKay-Navarro conjecture.
In this paper we propose a refinement of Sims conjecture concerning the cardinality of the point stabilizers in finite primitive groups and we make some progress towards this refinement. In this process, when dealing with primitive groups of diagonal type, we construct a finite primitive group $G$ on $Ω$ and two distinct points $α,β\in Ω$ with $G_{αβ}\unlhd G_α$ and $G_{αβ}\ne 1$, where $G_α$ is the stabilizer of $α$ in $G$ and $G_{αβ}$ is the stabilizer of $α$ and $β$ in $G$. In particular, this example gives an answer to a question raised independently by Peter Cameron and by Alexander Fomin…
Let $G$ be a finite group and $P$ a Sylow $2$-subgroup of $G$. We obtain both asymptotic and explicit bounds for the number of odd-degree irreducible complex representations of $G$ in terms of the size of the abelianization of $P$. To do so, we, on one hand, make use of the recent proof of the McKay conjecture for the prime 2 by Malle and Späth, and, on the other hand, prove lower bounds for the class number of the semidirect product of an odd-order group acting on an abelian $2$-group.
In this article, we study the elements with disconnected centralizer in the Brauer complex associated to a simple algebraic group G defined over a finite field with corresponding Frobenius map F and derive the number of F-stable semisimple classes of G with disconnected centralizer when the order of the fundamental group has prime order. We also discuss extendibility of semisimple characters to their inertia group in the full automorphism group. As a consequence, we prove that "twisted" and "untwisted" simple groups of type E_6 are "good" in defining characteristic, which is a contribution to …
