Related papers: The Navarro Conjecture for the alternating groups

Gamma-positivity appears frequently in finite geometries, combinatorics and number theory. Motivated by the recent work of Sagan and Tirrell (Adv. Math., 374 (2020), 107387), we study the relationships between gamma-positivity and…

A proof is given of several conjectures from a recent paper of Nakaya concerning the supersingular polynomial $ss_p^{(N*)}(X)$ for the Fricke group $\Gamma_0^*(N)$, for $N \in \{2, 3, 5, 7\}$. One of these conjectures gives a formula for…

A groupoid is alternative if it satisfies the alternative laws $x(xy)=(xx)y$ and $x(yy)=(xy)y$. These laws induce four partial maps on $\mathbb{N}^+\times \mathbb{N}^+$, $(r,\,s)\mapsto (2r,\,s-r)$, $(r-s,\,2s)$, $(r/2,\,s+r/2)$,…

Let X be a normal complex algebraic variety, and p a prime. We show that there exists an integer N=N(X, p) such that: any non-trivial, irreducible representation of the fundamental group of X, which arises from geometry, must be non-trivial…

Suppose given a Galois etale cover Y -> X of proper non-singular curves over an algebraically closed field k of prime characteristic p. Let H be its Galois group, and G any finite extension of H by a p-group P. We give necessary and…

In this paper, we show that each finite group $G$ containing at most $p^2$ Sylow $p$-subgroups for each odd prime number $p$, is a solvable group. In fact, we give a positive answer to the conjecture in \cite{Rob}.

The Evasiveness conjecture have been proved for properties of graphs on a prime-power number of vertices and the six vertices case. The ten vertices case is still unsolved. In this paper we study the size of the automorphism group of a…

Let $Q$ be a $p$-subgroup of a finite $p$-solvable group $G$, where $p$ is a prime, and suppose that $\delta$ is a linear character of $Q$ with the property that $\delta(x)=\delta(y)$ whenever $x,y\in Q$ are conjugate in $G$. In this…

We study the inverse problem in the difference Galois theory of linear differential equations over the difference-differential field $\mathbb{C}(x)$ with derivation $\frac{d}{dx}$ and endomorphism $f(x)\mapsto f(x+1)$. Our main result is…

We study the weight part of (a generalisation of) Serre's conjecture for mod l Galois representations associated to automorphic representations on rank two unitary groups for odd primes l. We propose a conjectural set of Serre weights,…

This article concerns the $p$-basic set existence problem in the representation theory of finite groups. We show that, for any odd prime $p$, the alternating group $\A_n$ has a $p$-basic set. More precisely, we prove that the symmetric…

The equivariant `main conjecture' of Iwasawa theory is shown to hold for a Galois extension $K/k$ of number fields with Galois group an $l$-adic pro-$l$ Lie group of dimension 1 containing an abelian subgroup of index $l$, provided that…

The McKay Conjecture (MC) asserts the existence of a bijection between the (inequivalent) complex irreducible representations of degree coprime to $p$ ($p$ a prime) of a finite group $G$ and those of the subgroup $N$, the normalizer of…

This note is a development of our two previous papers, arXiv:1212.3392v1 and 1306.3660v1. The fundamental question is whether there exists a Galois theory, in which the Galois group is a quantum group. For a linear equations with respect to…

A major difficult problem in Galois theory is the characterization of profinite groups which are realizable as absolute Galois groups of fields. Recently the Kernel $n$-Unipotent Conjecture and the Vanishing $n$-Massey Conjecture for $n\geq…

If G is a finite group, we have proposed new conjectures on the interaction between different primes and their corresponding Brauer principal blocks. In this paper, we give strong support to the validity of these conjectures.

In this paper we study the longstanding conjecture of whether there exists a noninner automorphism of order $p$ for a finite non-abelian $p$-group. We prove that if $G$ is a finite non-abelian $p$-group such that $G/Z(G)$ is powerful then…

In this note, I will prove a conjecture in \cite{BYY}, which is related to the invariants of a maximal parabolic subgroup of $\GL_{n+1}$. Consequently, the natural tensor invariants of this typical maximal parabolic subgroup of $\GL_{n+1}$…

We develop an abstract framework for studying the strong form of Malle's conjecture for nilpotent groups $G$ in their regular representation. This framework is then used to prove the strong form of Malle's conjecture for any nilpotent group…

Let $m$ be an integer greater than three and $\ell$ be an odd prime. In this paper, we prove that at least one of the following groups: $\mbox{P}\Omega^\pm_{2m}(\mathbb{F}_{\ell^s})$, $\mbox{PSO}^\pm_{2m}(\mathbb{F}_{\ell^s})$,…