Related papers: Notes on Loewy series of centers of $p$-blocks

Problem 21 of Brauer's list of problems from 1963 asks whether for any positive integer k there are finitely many isomorphism classes of groups that occur as the defect group of a block with k irreducible characters. We solve this problem…

We characterize the finite groups of minimal order that admit an irreducible complex character of degree $p$ or $p^2$, where $p$ is a prime.

Finite groups with very few character values are characterized. The following is the main result of this article: a finite non-abelian group has precisely four character values if and only if it is the generalized dihedral group of a…

We investigate the source algebra class of a p-block with cyclic defect groups of the group algebra of a finite group. By the work of Linckelmann this class is parametrized by the Brauer tree of the block together with a sign function on…

We show that the $p$-part of the degree of an irreducible character of a symmetric group is completely determined by the set of vanishing elements of $p$-power order. As a corollary we deduce that the set of zeros of prime power order…

This is a continuation of arXiv:0908.4053, where, among other things, we classified irreducible representations of the triplet vertex algebra W_{2,3}. In this part we extend the classification to W_{2,p}, for all odd p>3. We also determine…

It has been conjectured that if the number of distinct irreducible constituents of the product of two faithful irreducible characters of a finite $p$-group, for $p\geq5$, is bigger than $(p+1)/2$, then it is at least $p$. We give a…

Let $G$ be a finite $p$-solvable group, where $p$ is an odd prime. We establish a connection between extendible irreducible characters of subgroups of $G$ that lie under monomial characters of $G$ and nilpotent subgroups of $G$. We also…

We prove that if $p$ is an odd prime, $G$ is a solvable group, and the average value of the irreducible characters of $G$ whose degrees are not divisible by $p$ is strictly less than $2(p+1)/(p+3)$, then $G$ is $p$-nilpotent. We show that…

Let $U_l$ be a quantum group at an $l$th root of unity. Many tilting modules for $U_l$ have been shown to have what we call a balanced semisimple filtration, or a Loewy series whose semisimple layers are symmetric about some middle layer.…

Let $p$ be a prime such that $p \geq 5$. Let $G$ be a finite $p$-solvable group and let $p^a$ be the largest power of $p$ dividing $\chi(1)$ for an irreducible character $\chi$ of $G$, we show that $|G:F(G)|_p \leq p^{5.5a}$. Let $G$ be a…

Block character of a finite symmetric group S(n) is a positive definite function which depends only on the number of cycles in permutation. We describe the cone of block characters by identifying its extreme rays, and find relations of the…

Let $G$ be a finite group of order divisible by two distinct primes $p$ and $q$. We show that $G$ possesses a non-trivial irreducible character of degree not divisible by $p$ nor $q$ lying in both the principal $p$- and $q$-block whenever…

The decomposition matrix of a finite group in prime characteristic p records the multiplicities of its p-modular irreducible representations as composition factors of the reductions modulo p of its irreducible representations in…

This paper is concerned with the representation theory of finite groups. According to Robinson, the truth of certain variants of Alperin's weight conjecture on the $p$-blocks of a finite group would imply some arithmetical conditions on the…

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 $B$ be a $p$-block of a finite group, and set $m=$ $\sum \chi(1)^2$, the sum taken over all height zero characters of $B$. Motivated by a result of M. Isaacs characterising $p$-nilpotent finite groups in terms of character degrees, we…

As a generalization of the modular isomorphism problem we study the behavior of defect groups under Morita equivalence of blocks of finite groups over algebraically closed fields of positive characteristic. We prove that the Morita…

Let $G = {\rm U}(2m, {\mathbb F}_{q^2})$ be the finite unitary group, with $q$ the power of an odd prime $p$. We prove that the number of irreducible complex characters of $G$ with degree not divisible by $p$ and with Frobenius-Schur…

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…