32,103 research outputs found
Unipotent elements forcing irreducibility in linear algebraic groups
Let be a simple algebraic group over an algebraically closed field of
characteristic . We consider connected reductive subgroups of
that contain a given distinguished unipotent element of . A result of
Testerman and Zalesski (Proc. Amer. Math. Soc., 2013) shows that if is a
regular unipotent element, then cannot be contained in a proper parabolic
subgroup of . We generalize their result and show that if has order ,
then except for two known examples which occur in the case ,
the subgroup cannot be contained in a proper parabolic subgroup of . In
the case where has order , we also present further examples arising
from indecomposable tilting modules with quasi-minuscule highest weight.Comment: 33 page
Invariant forms on irreducible modules of simple algebraic groups
Let be a simple linear algebraic group over an algebraically closed field
of characteristic and let be an irreducible rational
-module with highest weight . When is self-dual, a basic
question to ask is whether has a non-degenerate -invariant alternating
bilinear form or a non-degenerate -invariant quadratic form.
If , the answer is well known and easily described in terms of
. In the case where , we know that if is self-dual, it
always has a non-degenerate -invariant alternating bilinear form. However,
determining when has a non-degenerate -invariant quadratic form is a
classical problem that still remains open. We solve the problem in the case
where is of classical type and is a fundamental highest weight
, and in the case where is of type and for . We also give a solution in some specific
cases when is of exceptional type.
As an application of our results, we refine Seitz's description of
maximal subgroups of simple algebraic groups of classical type. One consequence
of this is the following result. If are simple
algebraic groups and is irreducible, then one of the following
holds: (1) is not self-dual; (2) both or neither of the
modules and have a non-degenerate invariant
quadratic form; (3) , , and .Comment: 46 pages; to appear in J. Algebr
An almost sure ergodic theorem for quasistatic dynamical systems
We prove almost sure ergodic theorems for a class of systems called
quasistatic dynamical systems. These results are needed, because the usual
theorem due to Birkhoff does not apply in the absence of invariant measures. We
also introduce the concept of a physical family of measures for a quasistatic
dynamical system. These objects manifest themselves, for instance, in numerical
experiments. We then verify the conditions of the theorems and identify
physical families of measures for two concrete models, quasistatic expanding
systems and quasistatic dispersing billiards.Comment: 15 page
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