42 research outputs found
A brief guide to reversing and extended symmetries of dynamical systems
The reversing symmetry group is a well-studied extension of the symmetry
group of a dynamical system, the latter being defined by the action of a single
homeomorphism on a topological space. While it is traditionally considered in
nonlinear dynamics, where the space is simple but the map is complicated, it
has an interesting counterpart in symbolic dynamics, where the map is simple
but the space is not. Moreover, there is an interesting extension to the case
of higher-dimensional shifts, where a similar concept can be introduced via the
centraliser and the normaliser of the acting group in the full automorphism
group of the shift space. We recall the basic notions and review some of the
known results, in a fairly informal manner, to give a first impression of the
phenomena that can show up in the extension from the centraliser to the
normaliser, with some emphasis on recent developments.Comment: 18 pages, 2 figures; informal introduction and overvie
Discrete symmetries as automorphisms of the proper Poincare group
We present the consistent approach to finding the discrete transformations in
the representation spaces of the proper Poincar\'e group. To this end we use
the possibility to establish a correspondence between involutory automorphisms
of the proper Poincar\'e group and the discrete transformations. As a result,
we derive rules of the discrete transformations for arbitrary spin-tensor
fields without the use of relativistic wave equations. Besides, we construct
explicitly fields carrying representations of the extended Poincar\'e group,
which includes the discrete transformations as well.Comment: 33 pages, LaTe
Mackey's theory of -conjugate representations for finite groups. APPENDIX: On Some Gelfand Pairs and Commutative Association Schemes
The aim of the present paper is to expose two contributions of Mackey,
together with a more recent result of Kawanaka and Matsuyama, generalized by
Bump and Ginzburg, on the representation theory of a finite group equipped with
an involutory anti-automorphism (e.g. the anti-automorphism ).
Mackey's first contribution is a detailed version of the so-called Gelfand
criterion for weakly symmetric Gelfand pairs. Mackey's second contribution is a
characterization of simply reducible groups (a notion introduced by Wigner).
The other result is a twisted version of the Frobenius-Schur theorem, where
"twisted" refers to the above-mentioned involutory anti-automorphism. APPENDIX:
We consider a special condition related to Gelfand pairs. Namely, we call a
finite group and its automorphism satisfy Condition ()
if the following condition is satisfied: if for , and are conjugate in , then they are
conjugate in . We study the meanings of this condition, as well
as showing many examples of and which do (or do not) satisfy
Condition ().Comment: This consists of a 38 pages paper and a 7 pages APPENDIX. The
original version of the appendix appeared in the unofficial proceedings,
"Combinatorial Number Theory and Algebraic Combinatorics", November 18--21,
2002, Yamagata University, Yamagata, Japan, pp. 1--
Discrete symmetries as automorphisms of the proper Poincaré group
We present the consistent approach to finding the discrete transformations in the representation spaces of the proper Poincar\'e group. To this end we use the possibility to establish a correspondence between involutory automorphisms of the proper Poincar\'e group and the discrete transformations. As a result, we derive rules of the discrete transformations for arbitrary spin-tensor fields without the use of relativistic wave equations. Besides, we construct explicitly fields carrying representations of the extended Poincar\'e group, which includes the discrete transformations as well
Transitive Group Actions: (IM)PRIMITIVITY and Semiregular Subgroups
The following problem is considered: if is a semiregular abelian subgroup
of a transitive permutation group acting on a finite set , find
conditions for (non) existence of -invariant partitions of . Conditions
presented in this paper are derived by studying spectral properties of
associated -invariant digraphs. As an essential tool, irreducible complex
characters of are used.
Questions of this kind arise naturally when classifying combinatorial objects
which enjoy a certain degree of symmetry. As an illustration, a new and short
proof of an old result of Frucht, Graver and Watkins ({\it Proc. Camb. Phil.
Soc.}, {\bf 70} (1971), 211-218) classifying edge-transitive generalized
Petersen graphs, is given.Comment: 18 pages, 0 figure
On von Neumann's Examples of Types
The paper introduces in a new although maybe unusual form the examples of
types provided by J. von Neumann and F.J. Murray in their outstanding papers on
algebraic factorization (1936-1943)pursuing three main aims: speculating about
the physical reasons and motivations that are likely to have been at the origin
of von Neumann's investigation; describing the examples of factors provided by
those authors with the purpose of clarifying the general concepts standing at
the base of the classification of factors into three general types; outlining
the perspective of extending the theory to non--separable Hilbert spaces with
the purpose of suggesting a novel approach to the representation of infinite
systems controlled by external gauge fields.Comment: 24 pages, no figure
Homotopy approach to quantum gravity
I construct a finite-dimensional quantum theory from general relativity by a
homotopy method. Its quantum history is made up of at least two levels of
fermionic elements. Its unitary group has the diffeomorphism group as singular
limit. Its gravitational metrical form is the algebraic square. Its spinors are
multivectors.Comment: For International Journal of Theoretical Physics, Oberwolfach 2006
issu
Reflections and spinors on manifolds
This paper reviews some recent work on (s)pin structures and the Dirac
operator on hypersurfaces (in particular, on spheres), on real projective
spaces and quadrics. Two approaches to spinor fields on manifolds are compared.
The action of space and time reflections on spinors is discussed, also for
two-component (chiral) spinors.Comment: 10 pages, 1 figure, AMS-LaTe
Subrings of invariants for actions of finite dimensional Hopf algebras
This is a survey article on the invariant rings of Hopf actionsComment: plain Te
Introduction to Sporadic Groups for physicists
We describe the collection of finite simple groups, with a view on physical
applications. We recall first the prime cyclic groups , and the
alternating groups . After a quick revision of finite fields
, , with prime, we consider the 16 families of
finite simple groups of Lie type. There are also 26 \emph{extra} "sporadic"
groups, which gather in three interconnected "generations" (with 5+7+8 groups)
plus the Pariah groups (6). We point out a couple of physical applications,
including constructing the biggest sporadic group, the "Monster" group, with
close to elements from arguments of physics, and also the relation of
some Mathieu groups with compactification in string and M-theory.Comment: This paper is published in: Journal of Physics A, (vol.) 46, (2013),
as Topical Revie