1,013 research outputs found
Appendix A: Adequacy of representations of finite groups of Lie type
Thorne introduced the notion of adequate representations as a weakening of
the big representations used by Wiles and Taylor and others. In this appendix
to Dieulefait's paper, Automorphy of Symm5(GL(2)) and base change, we show that
certain representations of SL(2,q) are adequate. This is used by Dieulefait to
prove results about Hecke eigenforms of level 1 and newforms. We also prove
some general results about adequacy for representations of finite groups of Lie
type in the natural characteristic.Comment: This is appendix A to Dieulefait's paper, arXiv:1208.3946, mentioned
in the abstrac
On the singular value decomposition over finite fields and orbits of GU x GU
The singular value decomposition of a complex matrix is a fundamental concept
in linear algebra and has proved extremely useful in many subjects. It is less
clear what the situation is over a finite field. In this paper, we classify the
orbits of GU(m,q) x GU(n,q) on n by n matrices (which is the analog of the
singular value decomposition). The proof involves Kronecker's theory of pencils
and the Lang-Steinberg theorem for algebraic groups. Besides the motivation
mentioned above, this problem came up in a recent paper of Guralnick, Larsen
and Tiep where a concept of character level for the complex irreducible
characters of finite, general or special, linear and unitary groups was studied
and bounds on the number of orbits was needed. A consequence of this work
determines possible pairs of Jordan forms for nilpotent matrices of the form AB
where B is either the transpose of A or the conjugate transpose.Comment: 12 pages, second version has minor change
On the non-coprime k(GV) problem
Let V be a finite faithful completely reducible FG-module for a finite field
F and a finite group G. In various cases explicit linear bounds in |V| are
given for the numbers of conjugacy classes k(GV) and k(G) of the semidirect
product GV and of the group G respectively. These results concern the so-called
non-coprime k(GV)-problem.Comment: 26 page
On isometry groups of self-adjoint traceless and skew-symmetric matrices
This note is concerned with isometries on the spaces of self-adjoint
traceless matrices. We compute the group of isometries with respect to any
unitary similarity invariant norm. This completes and extends the result of
Nagy on Schatten -norm isometries. Furthermore, we point out that our proof
techniques could be applied to obtain an old result concerning isometries on
skew-symmetric matrices
Frobenius subgroups of free profinite products
We solve an open problem of Herfort and Ribes: Profinite Frobenius groups of
certain type do occur as closed subgroups of free profinite products of two
profinite groups. This also solves a question of Pop about prosolvable
subgroups of free profinite products.Comment: to appear in the Bulletin of the LM
Orders of Finite Groups of Matrices
We present a new proof of a theorem of Schur's determining the least common
multiple of the orders of all finite groups of complex -matrices
whose elements have traces in the field of rational numbers. The basic method
of proof goes back to Minkowski and proceeds by reduction to the case of finite
fields. For the most part, we work over an arbitrary number field rather than
the rationals. The first half of the article is expository and is intended to
be accessible to graduate students and advanced undergraduates. It gives a
self-contained treatment, following Schur, over the field of rational numbers
Essential dimension of algebraic groups, including bad characteristic
We give upper bounds on the essential dimension of (quasi-)simple algebraic
groups over an algebraically closed field that hold in all characteristics. The
results depend on showing that certain representations are generically free. In
particular, aside from the cases of spin and half-spin groups, we prove that
the essential dimension of a simple algebraic group of rank at least two is
at most . It is known that the
essential dimension of spin and half-spin groups grows exponentially in the
rank. In most cases, our bounds are as good or better than those known in
characteristic zero and the proofs are shorter. We also compute the generic
stabilizer of an adjoint group on its Lie algebra.Comment: v2 is a substantial revisio
Generically free representations II: irreducible representations
We determine which faithful irreducible representations of a simple
linear algebraic group are generically free for Lie(), i.e., which
have an open subset consisting of vectors whose stabilizer in Lie() is zero.
This relies on bounds on obtained in prior work (part I), which reduce
the problem to a finite number of possibilities for and highest weights for
, but still infinitely many characteristics. The remaining cases are handled
individually, some by computer calculation. These results were previously known
for fields of characteristic zero, although new phenomena appear in prime
characteristic; we provide a shorter proof that gives the result with very mild
hypotheses on the characteristic. (The few characteristics not treated here are
settled in part III.) These results are related to questions about invariants
and the existence of a stabilizer in general position.Comment: Part I is arxiv preprint 1711.05502. Part III is arxiv preprint
1801.06915. v2: minor text changes to align with part III; v3: updated to
align with v3 of Part I. Supporting Magma code available at
http://garibaldibros.co
Cosets of Sylow p-subgroups and a Question of Richard Taylor
We prove that for any prime p there exist infinitely many finite simple
groups G with a coset xP of a Sylow p-subgroup P of G such that every element
of xP has order divisible by p. John Thompson proved this for p=2 in 1967
answering a question of Lowell Paige. This result is used to answer a question
of Richard Taylor on adequate representations
Average dimension of fixed point spaces with applications
Let be a finite group, a field, and a finite dimensional
-module such that has no trivial composition factor on . Then the
arithmetic average dimension of the fixed point spaces of elements of on
is at most where is the smallest prime divisor of the
order of . This answers and generalizes a 1966 conjecture of Neumann which
also appeared in a paper of Neumann and Vaughan-Lee and also as a problem in
The Kourovka Notebook posted by Vaughan-Lee. Our result also generalizes a
recent theorem of Isaacs, Keller, Meierfrankenfeld, and Moret\'o. Various
applications are given. For example, another conjecture of Neumann and
Vaughan-Lee is proven and some results of Segal and Shalev are improved and/or
generalized concerning BFC groups
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