A reduction theorem for the McKay conjecture

The McKay conjecture asserts that for every finite group G and every prime p, the number of irreducible characters of G having p’-degree is equal to the number of such characters of the normalizer of a Sylow p-subgroup of G. Although this has been confirmed for large numbers of groups, including, for example, all solvable groups and all symmetric groups, no general proof has yet been found. In this paper, we reduce the McKay conjecture to a question about simple groups. We give a list of conditions that we hope all simple groups will satisfy, and we show that the McKay conjecture will hold for a finite group G if every simple group involved in G satisfies these conditions. Also, we establish that our conditions are satisfied for the simple groups PSL₂(q) for all prime powers q≥4, and for the Suzuki groups Sz(q) and Ree groups R(q), where q=2 ^e or q=3 ^e respectively, and e>1 is odd. Since our conditions are also satisfied by the sporadic simple group J ₁, it follows that the McKay conjecture holds (for all primes p) for every finite group having an abelian Sylow 2-subgroup.     

Inequalities for some blocks of finite groups

We analyze some inequalities for numbers of characters in p-blocks of finite groups and prove them in the case of blocks with a normal defect group, as well as in the case of certain blocks in non-abelian simple groups. Received: 27 October 2005     

Phase Diagrams of Semiconducting Compounds (III) Thermodynamic Calculation of the Ternary Melting Diagram of Ga?Al?As

The ternary phase diagram of Ga Al As has been calculated by VIELAND 'S method. The activity coefficients are determined using DARKEN 'S quadratic formalism for a ternary liquid in equilibrium with a regular solid solution. The agreement of calculation and experimental results is examinated using data from literature and solution measurements and liquid phase epitaxy experiments of this paper.     

The spin-1/2XXZ Heisenberg chain,the quantum algebra Uq[sl(2)], and duality transformations for minimal models

The finite-size scaling spectra of the spin-1/2XXZ Heisenberg chain with toroidal boundary conditions and an even number of sites provide a projection mechanism yielding the spectra of models with a central chargec < 1, including the unitary and nonunitary minimal series. Taking into account the half-integer angular momentum sectors—which correspond to chains with an odd number of sites—in many cases leads to new spinor operators appearing in the projected systems. These new sectors in theXXZ chain correspond to new types of frustration lines in the projected minimal models. The corresponding new boundary conditions in the Hamiltonian limit are investigated for the Ising model and the 3-state Potts model and are shown to be related to duality transformations which are an additional symmetry at their self-dual critical point. By different ways of projecting systems we find models with the same central charge sharing the same operator content and modular invariant partition function which, however, differ in the distribution of operators into sectors and hence in the physical meaning of the operators involved. Related to the projection mechanism in the continuum there are remarkable symmetry properties of the finiteXXZ chain. The observed degeneracies in the energy and momentum spectra are shown to be the consequence of intertwining relations involvingU _q [sl(2)] quantum algebra transformations.     

Topological Fixed Point Theorems Do Not Hold for Random Dynamical Systems

The aim of this paper is to demonstrate that topological fixed point theorems have no canonical generalization to the case of random dynamical systems. This is done by using tools from algebraic ergodic theory. We give a criterion for the existence of invariant probability measures for group valued cocycles. With that, examples of continuous random dynamical systems on a compact interval without random invariant points, which are an appropriate generalization of fixed points, are constructed.     

Quantum backaction of optical observations on Bose-Einstein condensates

Impressive pictures of moving Bose-Einstein condensates have been taken using phase-contrast imaging [M.R. Andrews et al., Science 273, 84 (1996)]. We calculate the quantum backaction of this measurement technique, assuming the absence of residual absorption. We find that the condensate gets gradually depleted at a universal rate that is proportional to the light intensity and to the inverse cube of the optical wave length. The fewer atoms are condensed the higher is the required intensity to see a picture, and, consequently, the higher is the induced backaction. To describe the quantum physics of phase-contrast imaging we put forward a new approach to quantum-optical propagation. We develop an effective field theory of paraxial optics in a fully quantized atomic medium. Received 25 February 1999