Related papers: The Character Triple Conjecture for maximal defect…

Schertz conjectured that every finite abelian extension of imaginary quadratic fields can be generated by the norm of the Siegel-Ramachandra invariants. We shall present a conditional proof of his conjecture by means of the characters on…

We show that almost every entry in the character table of $S_N$ is divisible by any fixed prime as $N\to\infty$. This proves a conjecture of Miller.

The Atiyah conjecture for a discrete group G states that the $L^2$-Betti numbers of a finite CW-complex with fundamental group G are integers if G is torsion-free and are rational with denominators determined by the finite subgroups of G in…

Let $H$ be a nonabelian finite simple group. Huppert's conjecture asserts that if $G$ is a finite group with the same set of complex character degrees as $H$, then $G\cong H\times A$ for some abelian group $A$. Over the past two decades,…

We prove \emph{the other half} of Brauer's Height Zero Conjecture in the case of principal blocks.

Suppose that $B$ is a Brauer $p$-block of a finite group $G$ with a unique modular character $\varphi$. We prove that $\varphi$ is liftable to an ordinary character of $G$ (which moreover is $p$-rational for odd $p$). This confirms the…

The so-called inductive McKay condition on finite simple groups, due to Isaacs-Malle-Navarro (2007), has been recently reformulated by Sp\"ath. We show that this reformulation applies to the reduction theorem for Alperin's weight…

We show that the refinement of Alperin's Conjecture proposed in "Frobenius Categories versus Brauer Blocks", Progress in Math. 274, can be proved by checking that this refinement holds on any central k*-extension of a finite group H…

We formulate an analogue of the conjecture of Birch and Swinnerton-Dyer for Abelian schemes with everywhere good reduction over higher dimensional bases over finite fields of characteristic $p$. We prove the prime-to-$p$ part conditionally…

In this paper, we calculate the numbers of irreducible ordinary characters and irreducible Brauer characters in a block of a finite group $G$, whose associated fusion system over a 2-subgroup $P$ of $G$ (which is a defect group of the…

Let p > 2 be prime. We state and prove (under mild hypotheses on the residual representation) a geometric refinement of the Breuil-M\'ezard conjecture for 2-dimensional mod p representations of the absolute Galois group of Qp. We also state…

In a recent paper, Gabriel Navarro and Pham Huu Tiep show that the so-called Alperin Weight Conjecture can be verified via the Classification of the Finite Simple Groups, provided any simple group fulfills a very precise list of conditions.…

Let $p$ be a prime and $G$ a finite group. A complex character of $G$ is called almost $p$-rational if its values belong to a cyclotomic field $\mathbb{Q}(e^{2\pi i/n})$ for some $n\in \mathbb{Z}^+$ prime to $p$ or precisely divisible by…

In this article we extend independent results of Lusztig and H\'ezard concerning the existence of irreducible characters of finite reductive groups, (defined in good characteristic and arising from simple algebraic groups), satisfying a…

We restrict the possibilities for the character degrees of $p$-groups $G$ satisfying $|G:G'| = p^2$. E.g. if $G$ is of maximal class and has an irreducible character of degree $> p$, then it has such a character of degree at most…

In this paper, we prove Lusztig's conjecture for finite special linear groups, i.e., we show that characteristic functions of character sheaves coincide with almost characters up to scalar constants, under the condition that the…

Let $G_2(q)$ be the Chevalley group of type $G_2$ defined over a finite field with q=p^n elements, where p is a prime number and $n$ is a positive integer. In this paper, we determine when the restriction of an absolutely irreducible…

We show that any triple Massey product with respect to prime 2 contains 0 whenever it is defined over any field. This extends the theorem of M. J. Hopkins and K. G. Wickelgren, from global fields to any fields. This is the first time when…

The classical Brauer-Siegel conjecture describes the asymptotic behaviour of the product of the class number and the regulator in families of number fields. All known cases of the conjecture rely on reducing the problem, via group theoretic…

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…