Principal blocks with six ordinary irreducible characters
We classify Sylow $p$-subgroups of finite groups whose principal $p$-blocks have precisely six ordinary irreducible characters.
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arXiv subjects
A. A. Schaeffer Fry
Publications and source records attributed to A. A. Schaeffer Fry.
On some rational extension properties for $GL_n(q)$ and even-degree characters fixed by order-2 Galois automorphisms
In this note, we prove that if every character of a finite group $G$ fixed by an order-2 Galois automorphism has odd degree, then $G$ has a normal Sylow $2$-subgroup. On the way, we study extensions of characters of $GL_n(q)$, $q$ odd, to the group extended by the transpose-inverse automorphism and prove that unipotent characters of $PSL_n(q)$ extend to rational characters of its automorphism group.
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The inductive McKay-Navarro conditions for the prime 2 and some groups of Lie type
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 called the inductive Galois--McKay) conditions hold for the prime $\ell = 2$ for several groups of Lie type, including the untwisted groups without nontrivial graph automorphisms.
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Galois Automorphisms and Classical Groups
In a previous work, the second-named author gave a complete description of the action of automorphisms on the ordinary irreducible characters of the finite symplectic groups. We generalise this in two directions. Firstly, using work of the first-named author, we give a complete description of the action of Galois automorphisms on irreducible characters. Secondly, we extend both descriptions to cover the case of special orthogonal groups. As a consequence, we obtain explicit descriptions for the character fields of symplectic and special orthogonal groups.
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Principal blocks for different primes, II
If G is a finite group, we have proposed new conjectures on the interaction between different primes and their corresponding Brauer principal blocks. In this paper, we give strong support to the validity of these conjectures.
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Principal blocks for different primes, I
We propose new conjectures about the relationship between the principal blocks of finite groups for different primes and establish evidence for these conjectures.
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Character degrees in blocks and defect groups
A recent question of Gabriel Navarro asks whether it is true that the derived length of a defect group is less than or equal to the number of degrees of irreducible characters in a block. In this article, we bring new evidence towards the validity of this statement.
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On Héthelyi-Külshammer's conjecture for principal blocks
We prove that the number of irreducible ordinary characters in the principal $p$-block of a finite group $G$ of order divisible by $p$ is always at least $2\sqrt{p-1}$. This confirms a conjecture of Héthelyi and Külshammer for principal blocks and provides an affirmative answer to Brauer's Problem 21 for principal blocks of bounded defect. Our proof relies on recent works of Maróti and Malle-Maróti on bounding the conjugacy class number and the number of $p'$-degree irreducible characters of finite groups, earlier works of Broué-Malle-Michel and Cabanes-Enguehard on the distribution of characters into unipotent blocks and $e$-Harish-Chandra series of finite reductive groups, and known cases of the Alperin-McKay conjecture.
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On the Inductive Alperin-McKay Conditions in the Maximally Split Case
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 $\mathbb{F}_q$ satisfy the inductive Alperin-McKay conditions for primes $\ell\geq 5$ and dividing $q-1$. Some methods to that end are adapted from the work of Malle--Späth.
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Galois-equivariant McKay bijections for primes dividing $q-1$
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.
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Groups with few $p'$-character degrees in the principal block
Let p be a prime larger than 3 and let G be a finite group. We prove that G is p-solvable of p-length at most 2 if there are at most two distinct character degrees relatively prime to p in the principal p-block of G. This generalizes a theorem of Isaacs-Smith, as well as a recent result of three of the present authors.
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Galois action on the principal block and cyclic Sylow subgroups
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.
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On the number of irreducible real-valued characters of a finite group
We prove that there exists an integer-valued function f on positive integers such that if a finite group G has at most k real-valued irreducible characters, then |G/Sol(G)| is at most f(k), where Sol(G) denotes the largest solvable normal subgroup of G. In the case k = 5, we further classify G/Sol(G). This partly answers a question of Iwasaki [15] on the relationship between the structure of a finite group and its number of real-valued irreducible characters.
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Galois Automorphisms on Harish-Chandra Series and Navarro's Self-Normalizing Sylow $2$-Subgroup Conjecture
G. Navarro has conjectured a necessary and sufficient condition for a finite group $G$ to have a self-normalizing Sylow 2-subgroup, which is given in terms of the ordinary irreducible characters of $G$. In a previous article, the author has reduced the proof of this conjecture to showing that certain related statements hold for simple groups. In this article, we describe the action of Galois automorphisms on the Howlett-Lehrer parametrization of Harish-Chandra induced characters. We use this to complete the proof of the conjecture by showing that the remaining simple groups satisfy the required conditions.
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Fields of Character Values for Finite Special Unitary Groups
Turull has described the fields of values for characters of $SL_n(q)$ in terms of the parametrization of the characters of $GL_n(q)$. In this article, we extend these results to the case of $SU_n(q)$.
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Principal $2$-Blocks and Sylow $2$-Subgroups
Let $G$ be a finite group with Sylow $2$-subgroup $P \leqslant G$. Navarro-Tiep-Vallejo have conjectured that the principal $2$-block of $N_G(P)$ contains exactly one irreducible Brauer character if and only if all odd-degree ordinary irreducible characters in the principal $2$-block of $G$ are fixed by a certain Galois automorphism $σ\in \mathrm{Gal}(\mathbb{Q}_{|G|}/\mathbb{Q})$. Recent work of Navarro-Vallejo has reduced this conjecture to a problem about finite simple groups. We show that their conjecture holds for all finite simple groups, thus establishing the conjecture for all finite groups.
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