Related papers: Character degrees in blocks and defect groups

Let \(G\) be a finite group, and let \(\Delta(G)\) denote the \emph{prime graph} built on the set of degrees of the irreducible complex characters of \(G\). It is well known that, whenever \(\Delta(G)\) is connected, the diameter of…

We propose upper bounds for the number of modular constituents of the restriction modulo $p$ of a complex irreducible character of a finite group, and for its decomposition numbers, in certain cases.

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…

Let $p$ be an odd prime. We show that for sufficiently large $n$, every $2$-block of $\mathfrak{S}_n$ and $\mathfrak{A}_n$ contains an ordinary irreducible character of degree divisible by $p$. For almost all $2$-blocks of $\mathfrak{A}_n$,…

Let G be a finite group. Denoting by cd(G) the set of degrees of the irreducible complex characters of G, we consider the character degree graph of G: this is the (simple undirected) graph whose vertices are the prime divisors of the…

Let $\chi$ be an irreducible character of a group $G.$ We denote the sum of the codegrees of the irreducible characters of $G$ by $S_c(G)=\sum_{\chi\in {\rm Irr}(G)}{\rm cod}(\chi).$ We consider the question if $S_c(G)\leq S_c(C_n)$ is true…

A numerical semigroup is an additive subsemigroup of the natural numbers that contains zero and has finite complement. A numerical semigroup is irreducible if it cannot be written as an intersection of numerical semigroups properly…

We give a classification, up to Morita equivalence, of 2-blocks of quasi-simple groups with abelian defect groups. As a consequence, we show that Donovan's conjecture holds for elementary abelian 2-groups, and that the entries of the Cartan…

We describe the ordinary characters of trivial source modules lying in blocks with cyclic defect groups relying on their recent classification in terms of paths on the Brauer tree by G.~Hiss and the second author. In particular, we show how…

We begin the study of character sheaves on a not necessarily connected reductive group, extending the known theory for connected groups.

We present some variations on some of the main open problems on character degrees. We collect some of the methods that have proven to be very useful to work on these problems. These methods are also useful to solve certain problems on zeros…

Answering a question of I. M. Isaacs, we show that the largest degree of irreducible complex representations of any finite non-abelian simple group can be bounded in terms of the smaller degrees. We also study the asymptotic behavior of…

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…

Let $G$ be a finite nilpotent group, $\chi$ and $\psi$ be irreducible complex characters of $G$ of prime degree. Assume that $\chi(1)=p$. Then either the product $\chi\psi$ is a multiple of an irreducible character or $\chi\psi$ is the…

We determine what are the fields of values of the irreducible $p$-height zero characters of all finite groups for $p=2$; we conjecture what they should be for odd primes, and reduce this statement to a problem on blocks of quasi-simple…

It is known that, if all the real-valued irreducible characters of a finite group have odd degree, then the group has normal Sylow $2$-subgroup. We generalize this result for Sylow $p$-subgroups, for any prime number $p$, while assuming the…

Let \(G\) be a finite solvable group, and let \(\Delta(G)\) denote the \emph{prime graph} built on the set of degrees of the irreducible complex characters of \(G\). A fundamental result by P.P. P\'alfy asserts that the complement…

In 1973, I. M. Isaacs described a correspondence between characters of degree not divisible by a fixed prime $p$ of a finite solvable group $G$ and those of the normalizer of Sylow $p$-subgroup of $G$, whenever the index of the normalizer…

Let $G$ be a finite group and let $p$ be a prime. Assume that there exists a prime $q$ dividing $|G|$ which does not divide the order of any $p$-local subgroup of $G$. If $G$ is $p$-solvable or $q$ divides $p-1$, then $G$ has a $p$-block of…

We consider graphs for which the non-backtracking matrix has defective eigenvalues, or graphs for which the matrix does not have a full set of eigenvectors. The existence of these values results in Jordan blocks of size greater than one,…