Irreducible subgroups of simple algebraic groups - a survey

Let $G$ be a simple linear algebraic group over an algebraically closed field $K$ of characteristic $p \geqslant 0$, let $H$ be a proper closed subgroup of $G$ and let $V$ be a nontrivial finite dimensional irreducible rational $KG$-module. We say that $(G,H,V)$ is an irreducible triple if $V$ is irreducible as a $KH$-module. Determining these triples is a fundamental problem in the representation theory of algebraic groups, which arises naturally in the study of the subgroup structure of classical groups. In the 1980s, Seitz and Testerman extended earlier work of Dynkin on connected subgroups in characteristic zero to all algebraically closed fields. In this article we will survey recent advances towards a classification of irreducible triples for all positive dimensional subgroups of simple algebraic groups.Comment: 31 pages; to appear in the Proceedings of Groups St Andrews 201

A note on the Zassenhaus product formula

We provide a simple method for the calculation of the terms c_n in the Zassenhaus product $e^{a+b}=e^a e^b \prod_{n=2}^{\infty} e^{c_n}$ for non-commuting a and b. This method has been implemented in a computer program. Furthermore, we formulate a conjecture on how to translate these results into nested commutators. This conjecture was checked up to order n=17 using a computer

A constructive algorithm for the Cartan decomposition of SU(2^N)

We present an explicit numerical method to obtain the Cartan-Khaneja-Glaser decomposition of a general element G of SU(2^N) in terms of its `Cartan' and `non-Cartan' components. This effectively factors G in terms of group elements that belong in SU(2^n) with n<N, a procedure that can be iterated down to n=2. We show that every step reduces to solving the zeros of a matrix polynomial, obtained by truncation of the Baker-Campbell-Hausdorff formula, numerically. All computational tasks involved are straightforward and the overall truncation errors are well under control.Comment: 15 pages, no figures, matlab file at http://cam.qubit.org/users/jiannis

Positive solutions of Schr\"odinger equations and fine regularity of boundary points

Given a Lipschitz domain $\Omega$ in ${\mathbb R} ^N$ and a nonnegative potential $V$ in $\Omega$ such that $V(x)\, d(x,\partial \Omega)^2$ is bounded in $\Omega$ we study the fine regularity of boundary points with respect to the Schr\"odinger operator $L_V:= \Delta -V$ in $\Omega$. Using potential theoretic methods, several conditions equivalent to the fine regularity of $z \in \partial \Omega$ are established. The main result is a simple (explicit if $\Omega$ is smooth) necessary and sufficient condition involving the size of $V$ for $z$ to be finely regular. An essential intermediate result consists in a majorization of $\int_A | {\frac {u} {d(.,\partial \Omega)}} | ^2\, dx$ for $u$ positive harmonic in $\Omega$ and $A \subset \Omega$. Conditions for almost everywhere regularity in a subset $A$ of $\partial \Omega$ are also given as well as an extension of the main results to a notion of fine ${\mathcal L}_1 | {\mathcal L}_0$-regularity, if ${\mathcal L}_j={\mathcal L}-V_j$, $V_0,\, V_1$ being two potentials, with $V_0 \leq V_1$ and ${\mathcal L}$ a second order elliptic operator.Comment: version 1. 23 pages version 3. 28 pages. Mainly a typo in Theorem 1.1 is correcte

Group-Theoretical Aspects of Orbifold and Conifold GUTs

Motivated by the simplicity and direct phenomenological applicability of field-theoretic orbifold constructions in the context of grand unification, we set out to survey the immensely rich group-theoretical possibilities open to this type of model building. In particular, we show how every maximal-rank, regular subgroup of a simple Lie group can be obtained by orbifolding and determine under which conditions rank reduction is possible. We investigate how standard model matter can arise from the higher-dimensional SUSY gauge multiplet. New model building options arise if, giving up the global orbifold construction, generic conical singularities and generic gauge twists associated with these singularities are considered. Viewed from the purely field-theoretic perspective, such models, which one might call conifold GUTs, require only a very mild relaxation of the constraints of orbifold model building. Our most interesting concrete examples include the breaking of E_7 to SU(5) and of E_8 to SU(4)xSU(2)xSU(2) (with extra factor groups), where three generations of standard model matter come from the gauge sector and the families are interrelated either by SU(3) R-symmetry or by an SU(3) flavour subgroup of the original gauge group.Comment: references adde

Discrete Symmetries from Broken SU(N)SU(N) and the MSSM

In order that discrete symmetries should not be violated by gravitational effects, it is necessary to gauge them. In this paper we discuss the gauging of $\Z_N$ from the breaking of a high energy $SU(N)$ gauge symmetry, and derive consistency conditions for the resulting discrete symmetry fr om the requirement of anomaly cancellation in the parent symmetry. These results are then applied to a detailed analysis of the possible discrete symmetries forbidding proton decay in the minimal supersymmetric standard model.Comment: 14 pages, plain TEX, computer problems fixed since first versio

A simple method for finite range decomposition of quadratic forms and Gaussian fields

We present a simple method to decompose the Green forms corresponding to a large class of interesting symmetric Dirichlet forms into integrals over symmetric positive semi-definite and finite range (properly supported) forms that are smoother than the original Green form. This result gives rise to multiscale decompositions of the associated Gaussian free fields into sums of independent smoother Gaussian fields with spatially localized correlations. Our method makes use of the finite propagation speed of the wave equation and Chebyshev polynomials. It improves several existing results and also gives simpler proofs.Comment: minor correction for t<

Manifolds with large isotropy groups

We classify all simply connected Riemannian manifolds whose isotropy groups act with cohomogeneity less than or equal to two.Comment: 21 page

Exact solutions in Einstein-Yang-Mills-Dirac systems

We present exact solutions in Einstein-Yang-Mills-Dirac theories with gauge groups SU(2) and SU(4) in Robertson-Walker space-time $R \times S^3$, which are symmetric under the action of the group SO(4) of spatial rotations. Our approach is based on the dimensional reduction method for gauge and gravitational fields and relates symmetric solutions in EYMD theory to certain solutions of an effective dynamical system. We interpret our solutions as cosmological solutions with an oscillating Yang-Mills field passing between topologically distinct vacua. The explicit form of the solution for spinor field shows that its energy changes the sign during the evolution of the Yang-Mills field from one vacuum to the other, which can be considered as production or annihilation of fermions. Among the obtained solutions there is also a static sphaleron-like solution, which is a cosmological analogue of the first Bartnik-McKinnon solution in the presence of fermions.Comment: 18 pages, LaTeX 2

Symmetric pairs and associated commuting varieties

We obtain a series of new results on the problem of irreducibility of commuting varieties associated with symmetric pairs or, in other words, $Z_2$-graded simple Lie algebras. In particular, we present many examples of reducible commuting varieties and show that the number of irreducible components can be arbitrarily large.Comment: 18 page