Symmetric group character degrees and hook numbers

In this article we prove the following result: for any two naturalnumbers k and , and for all sufficiently large symmetric groupsS_n, there are k disjoint sets of irreducible characters ofS_n, such that each set consists of characters with the samedegree, and distinct sets have different degrees. In particular,this resolves a conjecture most recently made by Moretóin [5]. The methods employed here are based upon the dualitybetween irreducible characters of the symmetric groups and thepartitions to which they correspond. Consequently, the paperis combinatorial in nature.     

Variations on average character degrees and p-nilpotence

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 there are examples that are not p-nilpotent where this bound is met for every prime p. We then prove a number of variations of this result.     

Derived lengths and character degrees

Let be a finite solvable group. Assume that the degree graph of has exactly two connected components that do not contain . Suppose that one of these connected components contains the subset , where and are coprime when . Then the derived length of is less than or equal to .

     

Retrieving information about a group from its character degrees or from its class sizes

We prove that knowledge of the character degrees of a finite group and of their multiplicities determines whether has a Sylow -subgroup as a direct factor. An analogous result based on knowledge of the conjugacy class sizes is known. We prove variations of both results and discuss their similarities.

     

A note on character degrees of finite solvable groups

In this paper we determine the structure of finite solvable groups with non-connected character degree graph and improve the related known results.     

Conjectures on the dilogarithm

The paper expresses Bloch's dilogarithm D(z) in terms of Clausen's function Cl₂(t). Using Lichtenbaum's conjecture on #K ₂ O _F for quadratic imaginary fields F, we get more than 50 conjectural linear relations between the values of Cl₂(t). We cannot prove most of them.     

On Isaacs' three character degrees theorem

Isaacs has proved that a finite group is solvable whenever there are at most three characters of pairwise distinct degrees in (Isaacs' three character degrees theorem). In this note, using Isaacs' result and the classification of the finite simple groups, we prove the solvability of whenever contains at most three monolithic characters of pairwise distinct degrees. §2 contains some additional results about monolithic characters.

     

The character sheaves on the group compactification

We give a definition of character sheaves on the group compactification which is equivalent to Lusztig's definition in [G. Lusztig, Parabolic character sheaves, II, Mosc. Math. J. 4 (4) (2004) 869-896]. We also prove some properties of the character sheaves on the group compactification.     

Cartan矩阵的分解和Brauer特征标次数猜想

设G为有限群,N是G的正规子群.记J=J(F[N])为F[N]的Jacobson根,I=Ann(J)={α∈F[G]|Jα=0}为J在F[G]中的零化子.本文主要研究了,根据F[G/N]和F[G]/I的Cartan矩阵,分解F[G]的Cartan矩阵.这种分解在Cartan不变量和G的合成因子之间建立了一些联系.本文指出N中p-亏零块的存在性依赖于Cartan不变量或者I在F[G]中的性质,证明了Cartan矩阵的分解部分地依赖于B所覆盖的N中的块的性质.本文研究了b为N上的块且l(b)=1时,覆盖b的G中的块B的性质.在两类情形下,本文证明了块代数上关于Brauer特征标次数的猜想成立,涵盖了Holm和Willems研究的某些情形.进而对Holm和Willems提出的问题给出了肯定的回答.另外,本文还给出了Cartan不变量的一些其它结果.     

A local conjecture on Brauer character degrees of finite groups

Recently, a new conjecture on the degrees of the irreducible Brauer characters of a finite group was presented by W. Willems. In this paper we propose a `local' version of this conjecture for blocks of finite groups, giving a lower bound for where the sum runs through the set of irreducible Brauer characters of in terms of invariants of . A slight reformulation leads to interesting open questions about traces of Cartan matrices of blocks.

We show that the local conjecture is true for blocks with one simple module, blocks of -solvable groups and blocks with cyclic defect groups. It also holds for many further examples of blocks of sporadic groups, symmetric groups or groups of Lie type. Finally we prove that the conjecture is true for blocks of tame representation type.

     

Finite groups with three composite character degrees

Let G be a finite group and let cd(G) be the set of all irreducible character degrees of G. We consider finite groups G with the property that cd(G) has at most three composite members. We derive a bound 8 for the size of character degree sets of such groups.