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

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

Green's and Dirichlet spaces associated with fine Markov processes

AbstractThis is the second paper in a series devoted to Green's and Dirichlet spaces. In the first paper, we have investigated Green's space K and the Dirichlet space H associated with a symmetric Markov transition function pt(x, B). Now we assume that p is a transition function of a fine Markov process X and we prove that: (a) the space H can be built from functions which are right continuous along almost all paths; (b) the positive cone K+ in K can be identified with a cone M of measures on the state space; (c) the positive cone H+ in H can be interpreted as the cone of Green's potentials of measures μ ϵ M. To every measurable set B in the state space E there correspond a subspace K(B) of K and a subspace H(B) of H. The orthogonal projections of K onto K and of H onto H(B) can be expressed in terms of the hitting probabilities of B by the Markov process X. As the main tool, we use additive functionals of X corresponding to measures μ ϵ M

Effects of strontium doping on oxygen reduction kinetics in thin-film LSCF cathodes

Thesis (M.S.)--Boston UniversityDense films of the mixed ionic-electronic conductor lanthanum strontium cobalt fenite (La1-xSrxCo0.2Fe0.8O3-δ) with x= 0.4, 0.3 and 0.2 (thereafter refened to as LSCF-6428, LSCF-7328 and LSCF-8228) were deposited in fixed patterns on yttria-stabilized zirconia (YSZ) substrates on top of a gadolinium-doped ceria (GDC) barrier layer by pulsed laser deposition (PLD) at the Pacific Northwest National Laboratory (PNNL) in Richland, WA. A counter electrode of porous 50-50 wt %/48.3-51.7 vol.% LCMIYSZ was screen-printed on the opposite side of the substrates. Electrochemical impedance spectroscopy (EIS) data were gathered for each of the compositions in air, but at varying temperatures (600, 700 and 800 °C). The total electrode polarization resistance, Rpob was plotted as a function of composition and temperature. Results show that, for all temperatures, the total polarization resistance drops considerably when the Sr composition is reduced from 0.4 to 0.3, and then increases slightly as the Sr composition is further reduced from 0.3 to 0.2 The relatively high polarization resistance for LSCF-6428 may be explained by recent evidence found by other members of our research group of surface strontium migration in LSCF-6428 films, which results in the formation of phases that could potentially affect electrochemical performance

Root Fernando-Kac subalgebras of finite type

Let $\mathfrak{g}$ be a finite-dimensional Lie algebra and $M$ be a $\mathfrak{g}$-module. The Fernando-Kac subalgebra of $\mathfrak{g}$ associated to $M$ is the subset $\mathfrak{g}[M]\subset\mathfrak{g}$ of all elements $g\in\mathfrak{g}$ which act locally finitely on $M$. A subalgebra $\mathfrak{l}\subset\mathfrak{g}$ for which there exists an irreducible module $M$ with $\mathfrak{g}[M]=\mathfrak{l}$ is called a Fernando-Kac subalgebra of $\mathfrak{g}$. A Fernando-Kac subalgebra of $\mathfrak{g}$ is of finite type if in addition $M$ can be chosen to have finite Jordan-H\"older $\mathfrak{l}$-multiplicities. Under the assumption that $\mathfrak{g}$ is simple, I. Penkov has conjectured an explicit combinatorial criterion describing all Fernando-Kac subalgebras of finite type which contain a Cartan subalgebra. In the present paper we prove this conjecture for $\mathfrak{g}\neq E_8$

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

Extending π\pi-systems to bases of root systems

Let $R$ be an indecomposable root system. It is well known that any root is part of a basis $B$ of $R$. But when can you extend a set of two or more roots to a basis $B$ of $R$? A $\pi$-system is a linearly independent set of roots, $C$, such that if $\alpha$ and $\beta$ are in $C$, then $\alpha - \beta$ is not a root. We will use results of Dynkin and Bourbaki to show that with two exceptions, $A_3 \subset B_n$ and $A_7 \subset E_8$, an indecomposable $\pi$-system whose Dynkin diagram is a subdiagram of the Dynkin diagram of $R$ can always be extended to a basis of $R$.Comment: 6 pages, LaTeX. Corrected typo in statement of theorem and clarified proo