arXiv subjects
Publications and source records attributed to Kay Magaard.
Motivated by the theory of Riemann surfaces and specifically the significance of Weierstrass points, we prove general structure results about finite groups that have a faithful transitive action with fixity 4. We also explain examples for many different possibilities of such actions.
As part of the programme to re-compute the character tables of all the groups in the Atlas we re-compute the character table of $\mathbb M$, the Monster simple group. We operate under the uniqueness hypotheses of $\mathbb M$ and the existence of an ordinary faithful representation of degree $196883 = 47.59.71$ and determine the conjugacy classes and centralizer orders of the elements of $\mathbb M$. Along the way we re-compute the character tables of centralizers of $p$-elements for $p < 11$ as well as fusions of conjugacy classes of these centralizers in $\mathbb M$.
We show the details of certain computations that are used in the paper "Verification of the conjugacy classes and ordinary character table of the Monster".
Motivated by the theory of Riemann surfaces and specifically the significance of Weierstrass points, we classify all finite simple groups that have a faithful transitive action with fixity 4, along with details about all possible such actions.
We determine the smallest irreducible Brauer characters for finite quasi-simple orthogonal type groups in non-defining characteristic. Under some restrictions on the characteristic we also prove a gap result showing that the next larger irreducible Brauer characters have a degree roughly the square of those of the smallest non-trivial characters.
We show the details of certain computations that are described in [BMW19].
We prove the correctness of the character table of the sporadic simple Baby Monster group that is shown in the Atlas of Finite Groups.
We develop in this work a method to parametrize the set $\mathrm{Irr}(U)$ of irreducible characters of a Sylow $p$-subgroup $U$ of a finite Chevalley group $G(p^f)$ which is valid for arbitrary primes $p$, in particular when $p$ is a very bad prime for $G$. As an application, we parametrize $\mathrm{Irr}(U)$ when $G=\mathrm{F}_4(2^f)$.
We parametrize the set of irreducible characters of the Sylow $p$-subgroups of the Chevalley groups $\mathrm{D}_6(q)$ and $\mathrm{E}_6(q)$, for an arbitrary power $q$ of any prime $p$. In particular, we establish that the parametrization is uniform for $p \ge 3$ in type $\mathrm{D}_6$ and for $p \ge 5$ in type $\mathrm{E}_6$, while the prime $2$ in type $\mathrm{D}_6$ and the primes $2,$ $3$ in type $\mathrm{E}_6$ yield character degrees of the form $q^m/p^i$ which force a departure from the generic situations. Also for the first time in our analysis we see a family of irreducible characters of a classical group of degree $q^m/p^i$ where $i > 1$ which occurs in type $\mathrm{D}_6$.
Let $q$ be a power of a prime $p$ and let $U(q)$ be a Sylow $p$-subgroup of a finite Chevalley group $G(q)$ defined over the field with $q$ elements. We first give a parametrization of the set $\text{Irr}(U(q))$ of irreducible characters of $U(q)$ when $G(q)$ is of type $\mathrm{G}_2$. This is uniform for primes $p \ge 5$, while the bad primes $p=2$ and $p=3$ have to be considered separately. We then use this result and the contribution of several authors to show a general result, namely that if $G(q)$ is any finite Chevalley group with $p$ a bad prime, then there exists a character $χ\in \text{Irr}(U(q))$ such that $χ(1)=q^n/p$ for some $n \in \mathbb{Z}_{\ge_0}$. In particular, for each $G(q)$ and every bad prime $p$, we construct a family of characters of such degree as inflation followed by an induction of linear characters of an abelian subquotient $V(q)$ of $U(q)$.
Let $U$ be a Sylow $p$-subgroup of the finite Chevalley group of type $D_4$ over the field of $q$ elements, where $q$ is a power of a prime $p$. We describe a construction of the generic character table of $U$.
This work completes the classification of the imprimitive irreducible modules, over algebraically closed fields of characteristic 0, of the finite quasisimple groups.
Let $q$ be a power of a prime $p$, let $G$ be a finite Chevalley group over $\mathbb{F}_q$ and let $U$ be a Sylow $p$-subgroup of $G$; we assume that $p$ is not a very bad prime for $G$. We explain a procedure of reduction of irreducible complex characters of $U$, which leads to an algorithm whose goal is to obtain a parametrization of the irreducible characters of $U$ along with a means to construct these characters as induced characters. A focus in this paper is determining the parametrization when $G$ is of type $\mathrm{F}_4$, where we observe that the parametrization is "uniform" over good primes $p > 3$, but differs for the bad prime $p = 3$. We also explain how it has been applied for all groups of rank $4$ or less.
Let $UY_n(q)$ be a Sylow p-subgroup of an untwisted Chevalley group $Y_n(q)$ of rank n defined over $\mathbb{F}_q$ where q is a power of a prime p. We partition the set $Irr(UY_n(q))$ of irreducible characters of $UY_n(q)$ into families indexed by antichains of positive roots of the root system of type $Y_n$. We focus our attention on the families of characters of $UY_n(q)$ which are indexed by antichains of length 1. Then for each positive root $α$ we establish a one to one correspondence between the minimal degree members of the family indexed by $α$ and the linear characters of a certain subquotient $\overline{T}_α$ of $UY_n(q)$. For $Y_n = A_n$ our single root character construction recovers amongst other things the elementary supercharacters of these groups. Most importantly though this paper lays the groundwork for our classification of the elements of $Irr(UE_i(q))$, $6 \le i \le 8$ and $Irr(UF_4(q))$.
In this paper we analyze the structure of transitive permutation groups that have trivial four point stabilizers, but some nontrivial three point stabilizer. In particular we give a complete, detailed classification when the group is simple or quasisimple. This paper is motivated by questions concerning the relationship between fixed points of automorphisms of Riemann surfaces and Weierstrass points and is a continuation of the authors' earlier work.
We construct extensions of the field of rational numbers with the Galois group G_2(F_p) by reducing p-adic representations attached to automorphic representations.
We show that if the action of a classical group $G$ on a set $Ω$ of $1$-spaces of its natural module is of genus at most two, then $|Ω| \leq 10,000$.
We determine the image of the braid groups inside the Iwahori-Hecke algebras of type A, when defined over a finite field, in the semisimple case, and for suitably large (but controlable) order of the defining (quantum) parameter.