We develop the concept of character level for the complex irreducible characters of finite, general or special, linear and unitary groups. We give characterizations of the level of a character in terms of its Lusztig label and in terms of its degree. Then we prove explicit upper bounds for character values at elements with not-too-large centralizers and derive upper bounds on the covering number and mixing time of random walks corresponding to these conjugacy classes. We also characterize the level of the character in terms of certain dual pairs and prove explicit exponential character bounds for the character values, provided that the level is not too large. Several further applications are also provided. Related results for other finite classical groups are obtained in the sequel [Guralnick et al. ‘Character levels and character bounds for finite classical groups’, Preprint, 2019, arXiv:1904.08070] by different methods.

We prove that spin groups act generically freely on various spinor modules, in the sense of group schemes and in a way that does not depend on the characteristic of the base field. As a consequence, we extend the surprising calculation of the essential dimension of spin groups and half-spin groups in characteristic zero by Brosnan et al. [Essential dimension, spinor groups, and quadratic forms, Ann. of Math. (2) 171 (2010), 533–544], and Chernousov and Merkurjev [Essential dimension of spinor and Clifford groups, Algebra Number Theory 8 (2014), 457–472] to fields of characteristic different from two. We also complete the determination of generic stabilizers in spin and half-spin groups of low rank.

Extant species of the genus Equus (e.g., horses, asses, and zebras) have a widespread distribution today on all continents except Antarctica. Extinct species of Equus represented by fossils were likewise widely distributed in the Pliocene and even more so during the Pleistocene. In order to understand the efficacy of “big data” for (paleo)biogeographic analyses, location records (latitude, longitude) and fossil occurrences for the genus Equus were mined and further explored from six databases, including iDigBio, Paleobiology Database, VertNet, BISON, Neotoma, and GBIF. These were chosen from a priori knowledge of where relevant data might be aggregated. We also realized that these databases have different objectives and data sources and therefore would provide a useful comparative study of the widespread taxon Equus in space and time.

The mining of Equus data from these six sources yielded a combined total of 123.8 K location records, including 116.2K fossil specimens. These include individual points that are unique, that is, only occurring in one of these databases, and those that are duplicated in multiple databases. Of the six databases, three (iDigBio, Paleobiology Database, and GBIF) were judged to be the most useful in the Equus use case. Most of the databases are biased toward North American records, thus limiting the reconstruction of the actual distribution of the genus Equus in space and time outside of this continent. Although Equus has a large number of digitally accessible records, fundamentally interesting questions pertaining to evolutionary dynamics and extinction geography are still a challenge for these kinds of biodiversity databases due primarily to the lack of sufficiently dense and precise temporal data.

Understanding the response of a species or lineage to long-term environmental change is a critical aspect of evolutionary paleoecology. In order to do this, paleobiologists must have an excellent fossil record of a lineage and an independent source of environmental data in the same region. This situation occurs in the San Pedro area of southern California, where relatively new paleotemperature and paleoproductivity records enhance the well-known fossil gastropod record. We quantified shell morphology of late Pleistocene and Recent turritellid gastropods from this area and compared the timing of changes with temperature and productivity reconstructions for the region. Our results indicate that warm temperatures and moderate to high productivity are associated with larger shells and wider whorls. Cold temperatures and lower productivity are associated with smaller, narrower shells. We propose that warm temperatures and moderate productivity result in higher growth rates in turritellid gastropods. Our work also suggests that below a certain threshold temperature, productivity appears to have no influence on shell morphology. In other words, growth rate is unaffected by high productivity unless average temperatures are above a certain level. These results are consistent with models of shell deposition and with experimental results from living gastropods and bivalves reported in the literature.

We study the problem of determining, for a polynomial function $f$ on a vector space $V$, the linear transformations $g$ of $V$ such that $f\circ g=f$. When $f$ is invariant under a simple algebraic group $G$ acting irreducibly on $V$, we note that the subgroup of $\text{GL}(V)$ stabilizing $f$ often has identity component $G$, and we give applications realizing various groups, including the largest exceptional group $E_{8}$, as automorphism groups of polynomials and algebras. We show that, starting with a simple group $G$ and an irreducible representation $V$, one can almost always find an $f$ whose stabilizer has identity component $G$, and that no such $f$ exists in the short list of excluded cases. This relies on our core technical result, the enumeration of inclusions $G<H\leqslant \text{SL}(V)$ such that $V/H$ has the same dimension as $V/G$. The main results of this paper are new even in the special case where $k$ is the complex numbers.

We prove the existence of certain rationally rigid triples in ${E}_{8}(p)$ for good primes $p$ (i.e. $p>5$) thereby showing that these groups occur as Galois groups over the field of rational numbers. We show that these triples arise from rigid triples in the algebraic group and prove that they generate an interesting subgroup in characteristic zero. As a byproduct of the proof, we derive a remarkable symmetry between the character table of a finite reductive group and that of its dual group. We also give a short list of possible overgroups of regular unipotent elements in simple exceptional groups.

Perhaps unexpectedly, there is a rich and deep connection between field of values of characters, their degrees and the structure of a finite group. Some of the fundamental results on the degrees of characters of finite groups, as the Ito–Michler and Thompson's theorems, admit a version involving only characters with certain fixed field of values ([DNT, NS, NST2, NT1, NT3]).

We study Frobenius groups acting on curves.

The low-dimensional projective irreducible representations in cross characteristics of the projective special linear group $\mbox{PSL}_{n}(q)$ are investigated.

If $n \geq 3$ and $(n,q) \neq (3,2)$, $(3,4)$, $(4,2)$, $(4,3)$, all such representations of the first degree (which is $(q^{n}-q)/(q-1) - \kappa$ with $\kappa = 0$ or $1$) and the second degree (which is $(q^{n}-1)/(q-1)$) come from Weil representations. We show that the gap between the second and the third degree is roughly $q^{2n-4}$.

In this paper we obtain a classification of those subgroups of the finite general linear group $\mbox{GL}_d (q)$ with orders divisible by a primitive prime divisor of $q^e - 1$ for some $e > \frac{1}{2}d$. In the course of the analysis, we obtain new results on modular representations of finite almost simple groups. In particular, in the last section, we obtain substantial extensions of the results of Landazuri and Seitz on small cross-characteristic representations of some of the finite classical groups.

In this article, we obtain results on commutators in Sylow subgroups of the lower central series, extending the work of Dark and Newell [2], Rodney [12, 13] and Aschbacher and the author [1, 6, 7].

Some notation is required for the statement of the main results. Let r be a positive integer and define

and

where x1, …, xr, are elements in a group G. Let ΓrG = {[x1, …, xr]∣ x1 ∈ G} be the set of r-fold commutators in G. Then Lr,G = 〈ΓrG〉 is the rth term in the lower central series of G. Set L∞G = ∩ Lr,G.

Let G be a group. The rth term LrG of the lower central series of G is the subgroup generated by the r-fold commutators

where , and for r > 1,

Let K be an algebraically closed field of characteristic zero, and S a nonempty subset of K such that S Q = Ø and card S < card K, where Q is the field of rational numbers. By Zorn's Lemma, there exist subfields F of K which are maximal with respect to the property of being disjoint from S. This paper examines such subfields and investigates the Galois group Gal K/F along with the lattice of intermediate subfields.

Let G be a group. We define λ(G) to be the smallest integer n such that every element of the commutator subgroup G′ is a product of n commutators. Ito [4] has shown that λ (An)= 1 for all n. Thompson [7] has shown that λ (SLn(q))= 1 for all n and q. In fact, there is no known simple group G such that λ(G)>1. However, there do exist such perfect groups (cf. [7]).