arXiv subjects
Publications and source records attributed to Frank Lübeck.
In this short note we show that every connected reductive simply-connected algebraic group of rank $>1$ over the complex numbers has infinitely many pairs of irreducible representations which are not related by an automorphism of the algebraic group and which have the same degree. This answers a question I was asked by J.~P.~Serre.
The values of the ordinary Green functions are known for almost all groups of Lie type, a long term achievement by various authors. In this note we solve the last open cases, which are for exceptional groups $E_8(q)$ where $q$ is a power of $2$, $3$ or $5$.
Using computational methods, we determine the irreducible Brauer characters of the automorphism group of the Chevalley group F_4(2), up to one parameter and one consistency issue.
In this paper, we formulate the notion of split elements of a unipotent class in a connected reductive group $G$. Generalized Green functions of $G$ can be computed by using Lusztig's algorithm, if split elements exist for any unipotent class. The existence of split elements is reduced to the case where $G$ is a simply connected, almost simple group. We show, in the case of classical groups, split elements exist, which is a refinement of previous results. In the case of exceptional groups, we show the existence of split elements, possibly except one class for $G$ of type $E_7$.
We describe how to compute for two polynomials $f(X), g(X) \in \mathbb{Z}[X]$ with integer coefficients the greatest common divisors of $f(z)$ and $g(z)$ for all integers $z \in \mathbb{Z}$. As an application we determine the structures (elementary divisors) of all maximal tori in exceptional groups of Lie type.
We define standardized constructions of finite fields, and standardized generators of (multiplicative) cyclic subgroups in these fields. The motivation is to provide a substitute for Conway polynomials which can be used by various software packages and in collections of mathematical data which involve finite fields.
We verify the inductive blockwise Alperin weight condition in odd characteristic $\ell$ for the finite exceptional Chevalley groups $F_4(q)$ for $q$ not divisible by $\ell$.
Groups of structure $2.O_8^+(2)$ have an irreducible representation of degree $8$ which can be realized over $\mathbb{Z}$ and any prime field $\mathbb{F}_p$. This representation extends to a group of structure $2.O_8^+(2).2$. Any subgroup $Z \leq \mathbb{F}_p^{\times}$ acts by scalar multiplication on this module over $\mathbb{F}_p$. In this short note we determine for which primes $p > 7$ and which $Z$ the central products $Z \circ (2.O_8^+(2)$ and $Z \circ (2.O_8^+(2).2)$ have a regular orbit on the $8$-dimensional $\mathbb{F}_p$-module. This work was triggered by an omission in the paper by Köhler and Pahlings with title 'Regular Orbits and the $k(GV)$-Problem', a paper which is used in various places in work on the $k(GV)$-problem.
We describe the computation of generalized Green functions and 2-parameter Green functions for finite reductive groups.
We relate character theory of the symmetric groups $S_{2n}$ and $S_{2n+1}$ with that of the hyperoctahedral group $B_n = ({\mathbb Z}/2)^n \rtimes S_n$, as part of the expectation that the character theory of reductive groups with diagram automorphism and their Weyl groups, is related to the character theory of the fixed subgroup of the diagram automorphism.
Using computational methods, we complete the determination of the $3$-modular character table of the Chevalley group $F_4(2)$ and its covering group.
Let $G$ be a simple algebraic group over the algebraic closure of $GF(p)$ ($p$ prime), and let $G(q)$ denote a corresponding finite group of Lie type over $GF(q)$, where $q$ is a power of $p$. Let $X$ be an irreducible subvariety of $G^r$ for some $r\ge 2$. We prove a zero-one law for the probability that $G(q)$ is generated by a random $r$-tuple in $X(q) = X\cap G(q)^r$: the limit of this probability as $q$ increases (through values of $q$ for which $X$ is stable under the Frobenius morphism defining $G(q)$) is either 1 or 0. Indeed, to ensure that this limit is 1, one only needs $G(q)$ to be generated by an $r$-tuple in $X(q)$ for two sufficiently large values of $q$. We also prove a version of this result where the underlying characteristic is allowed to vary. In our main application, we apply these results to the case where $r=2$ and the irreducible subvariety $X = C\times D$, a product of two conjugacy classes of elements of finite order in $G$. This leads to new results on random $(2,3)$-generation of finite simple groups $G(q)$ of exceptional Lie type: provided $G(q)$ is not a Suzuki group, we show that the probability that a random involution and a random element of order 3 generate $G(q)$ tends to $1$ as $q \rightarrow \infty$. Combining this with previous results for classical groups, this shows that finite simple groups (apart from Suzuki groups and $PSp_4(q)$) are randomly $(2,3)$-generated. Our tools include algebraic geometry, representation theory of algebraic groups, and character theory of finite groups of Lie type.
We describe two methods for computing $p$-modular Brauer character tables for groups of Lie type $G(p^f)$ in defining characteristic $p$, assuming that the ordinary character table of $G(p^f)$ is known, and the weight multiplicities of the corresponding algebraic group $G$ are known for $p$-restricted highest weights.
We discuss a practical algorithm to compute parabolic Kazhdan-Lusztig polynomials. As an application we compute Kazhdan-Lusztig polynomials which are needed to evaluate a character formula for reductive groups due to Lusztig. Some coefficients of these polynomials have interesting interpretations for certain finite groups. We find examples of finite dimensional modules for finite groups with much higher dimensional first cohomology group than in all previously known cases. Some of these examples lead to the construction of finite groups with many maximal subgroups, contradicting an old conjecture by G.~E.~Wall.
We derive a Murnaghan--Nakayama type formula for the values of unipotent characters of finite classical groups on regular semisimple elements. This relies on Asai's explicit decomposition of Lusztig restriction. We use our formula to show that most complex irreducible characters vanish on some $\ell$-singular element for certain primes $\ell$. As an application we classify the simple endotrivial modules of the finite quasi-simple classical groups. As a further application we show that for finite simple classical groups and primes $\ell\ge3$ the first Cartan invariant in the principal $\ell$-block is larger than~2 unless Sylow $\ell$-subgroups are cyclic.
Primitive prime divisors play an important role in group theory and number theory. We study a certain number theoretic quantity, called $Φ^*_n(q)$, which is closely related to the cyclotomic polynomial $Φ_n(x)$ and to primitive prime divisors of $q^n-1$. Our definition of $Φ^*_n(q)$ is novel, and we prove it is equivalent to the definition given by Hering. Given positive constants $c$ and $k$, we give an algorithm for determining all pairs $(n,q)$ with $Φ^*_n(q)\le cn^k$. This algorithm is used to extend (and correct) a result of Hering which is useful for classifying certain families of subgroups of finite linear groups.
Let $G$ be the finite simple Chevalley group of type $^2E_6(2)$. It has a Schur multiplier of type $C_2^2 \times C_3$. We determine the ordinary character tables of the central extensions $3.G$, $6.G$, $(2^2\times 3).G$ of $G$ and their extensions by an automorphism of order $2$, that is $3.G.2$, $6.G.2$ and $(2^2\times 3).G.2$. Furthermore we determine all Brauer trees of all groups of type $Z.G.A$ (where $Z$ is central in $Z.G \lhd Z.G.A$ and $A \cong Z.G.A/Z.G$) for which the ordinary character table is known.
We prove the existence of certain rationally rigid triples in F_4(p) for good primes p (i.e., p>3), thereby showing that these groups occur as regular Galois groups over Q(t) and so also over Q. We show that these triples give rise to rigid triples in the algebraic group and prove that they generate an interesting subgroup in characteristic 0.