Related papers: Minimal heights and defect groups with two charact…

Eaton and Moret\'o proposed an extension of Brauer's famous height zero conjecture on blocks of finite groups to the case of non-abelian defect groups, which predicts the smallest non-zero height in such blocks in terms of local data. We…

The Eaton--Moret\'o conjecture extends the recently-proven Brauer height zero conjecture to blocks with non-abelian defect group, positing equality between the minimal positive heights of a block of a finite group and its defect group. Here…

Recently, a new conjecture on the degrees of the irreducible Brauer characters of a finite group was presented by the second author. In this paper we propose a 'local' version of this conjecture for blocks B of finite groups, giving a lower…

We prove that Brauer's Height Zero Conjecture holds for p-blocks of finite groups with metacyclic defect groups. If the defect group is nonabelian and contains a cyclic maximal subgroup, we obtain the distribution into p-conjugate and…

We propose a projective version of the celebrated Brauer's Height Zero Conjecture on characters of finite groups and prove it, among other cases, for $p$-solvable groups as well as for (some) quasi-simple groups.

Let $B$ be a $p$-block of a finite group $G$ with defect group $D$. The more difficult direction of the recently proven height zero conjecture says that $D$ is abelian if every character in Irr$(B)$ has height zero. We consider a smaller…

We prove, for primes $p\ge5$, two inequalities between the fundamental invariants of Brauer $p$-blocks of finite quasi-simple groups: the number of characters in the block, the number of modular characters, the number of height zero…

We show that a conjecture of Giannelli on character degrees of height zero characters holds for blocks with a cyclic or Klein four defect group.

We prove that Sp\"ath's Character Triple Conjecture holds for every finite group with respect to maximal defect characters at the prime 2. This is done by reducing the maximal defect case of the conjecture to the so-called inductive…

Several recent problems in the representation theory of finite groups require determining whether certain characters of almost simple groups belong to the principal block. Since the values of these characters are not yet known, we employ…

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…

Let G be a finite group, let N be a normal subgroup of G, and let theta in Irr(N) be a G-invariant character. We fix a prime p, and we introduce a canonical partition of Irr(G|theta) relative to p. We call each member B_theta of this…

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…

Let \chi be an irreducible character of the finite group G. If g is an element of G and \chi(g) is not zero, then we conjecture that the order of g divides |G|/\chi(1). The conjecture is a generalization of the classical fact that…

Suppose that $B$ is a Brauer $p$-block with defect group $D$. If $B$ exactly contains 4 irreducible characters, then we show that $D$ has order 4 or 5, assuming the Alperin--McKay conjecture.

In this note we give applications of recent results coming mostly from the third paper of this series. It is shown that the number of irreducible characters in a $p$-block of a finite group with abelian defect group $D$ is bounded by $|D|$…

We prove a strengthening of Brauer's height zero conjecture for principal 2-blocks with Galois automorphisms. This requires a new extension of the It\^o--Michler theorem for the prime~2, again with Galois automorphisms. We close, this time…

We study Brauer's long-standing $k(B)$-conjecture on the number of characters in $p$-blocks for finite quasi-simple groups and show that their blocks do not occur as a minimal counterexample for $p\ge5$ nor in the case of abelian defect.…

We determine the smallest irreducible Brauer characters for finite quasi-simple orthogonal type groups in non-defining characteristic. Under some restrictions on the characteristic we also prove a gap result showing that the next larger…

Let $G$ be a finite group and let $p$ be a prime. In this paper, we prove a strengthened version of Brauer's height zero conjecture for the principal $p$-block of $G$ that takes the action of a certain group of Galois automorphisms into…