Related papers: Groups where all the irreducible characters are su…

A super-Brauer character theory of a group $G$ and a prime $p$ is a pair consisting of a partition of the irreducible $p$-Brauer characters and a partition of the $p$-regular elements of $G$ that satisfy certain properties. We classify the…

It is well-known that the representation theory of the finite group of unipotent upper-triangular matrices $U_n$ over a finite field is a wild problem. By instead considering approximately irreducible representations (supercharacters), one…

The purpose of the present paper is to investigate a hypergroup arising from irreducible characters of a compact group G and a closed subgroup of G with finite index. The convolution of this hypergroup is introduced by inducing irreducible…

Let $P$ be a Sylow $p$-subgroup of a finite $p$-solvable group $G$, where $p$ is a prime. Using a normal $p$-series $\mathcal{N}$ of $G$, we introduce the notion of $(\mathcal{N},p)$-stable characters and prove that $G$ and ${\bf N}_G(P)$…

Let $G$ be a finite group. Denote by $\textrm{Irr}(G)$ the set of all irreducible complex characters of $G.$ Let $\textrm{cd}(G)=\{\chi(1)\;|\;\chi\in \textrm{Irr}(G)\}$ be the set of all irreducible complex character degrees of $G$…

We investigate a beautiful conjecture of T. Wilde on character values and element orders of finite groups. We reduce it to a statement on nearly simple groups that can be checked ``prime by prime". For these groups, we show that a strong…

We prove that an abstract (possibly infinite dimensional) complex irreducible representation of a discrete supersolvable group is monomial if and only if it has finite weight. We also prove a general result that implies converse of Schur's…

We construct supercharacter theories of finite unipotent groups in the orthogonal, symplectic and unitary types. Our method utilizes group actions in a manner analogous to that of Diaconis and Isaacs in their construction of supercharacters…

The notion of a supercharacter theory was proposed by P. Diaconis and I.M. Isaacs in 2008. A supercharacter theory for a given finite group is a pair of the system of certain complex characters and the partition of group into classes that…

We define and study supercharacters of the classical finite unipotent groups of symplectic and orthogonal types (over any finite field of odd characteristic). We show how supercharacters for groups of those types can be obtained by…

In representation theory of finite groups an important role is played by irreducible characters of p-defect 0, for a prime p dividing the group order. These are exactly those vanishing at the p-singular elements. In this paper we generalize…

Let F be a finite field with q elements, let A be a finite dimensional F-algebra and let J=J(A) be the Jacobson radical of A. Then G=1+J is a p-group, where p is the characteristic of F. We refer to G as an F-algebra group. A subgroup H of…

We define the superclasses for a classical finite unipotent group $U$ of type $B_{n}(q)$, $C_{n}(q)$, or $D_{n}(q)$, and show that, together with the supercharacters defined in a previous paper, they form a supercharacter theory. In…

We prove that, in a finite group, if every rational irreducible character has odd degree, then all rational elements are 2-elements, as it was originally conjectured by Tiep and Tong-Viet.

We extend the Howlett-Isaacs theorem on the solvability of groups of central type taking into account actions by automorphisms. Then we study certain induced characters whose constituents have all the same degree.

For a prime $p$, we determine a Sylow $p$-subgroup $D$ of a finite group $G$ such that the principal $p$-block $B$ of $G$ has four irreducible ordinary characters. It has been determined already for the cases where the number is up to three…

A result of D. Segal states that every complex irreducible representation of a finitely generated nilpotent group $G$ is monomial if and only if $G$ is abelian-by-finite. A conjecture of A. N. Parshin, recently proved affirmatively by I.V.…

Let $q$ be a prime power and $U$ the group of lower unitriangular matrices of order $n$ for some natural number $n$. We give a lower bound for the degrees of irreducible constituents of Andr\'{e}-Yan supercharacters and classify the…

We prove that the order of a finite group $G$ with trivial solvable radical is bounded above in terms of ${\rm acd}(G)$, the average degree of the irreducible characters. It is not true that the index of the Fitting subgroup is bounded…

The classification of irreducible, spherical characters of the infinite-dimensional unitary/orthogonal/symplectic groups can be obtained by finding all possible limits of normalized, irreducible characters of the corresponding…