Dice and Divine Action

The world picture disclosed by classical physics, beginning with Kepler, Galileo, and Newton and continuing until the early part of the twentieth century, was increasingly deterministic. The prevailing metaphor was that of a clock: as a clock is wound up and runs in an entirely predictable way which can be known from its initial conditions, so the universe unfolds according to a deterministic plan. Such a universe, unfolding under the influence of forces so absolute they were called ‘laws’, appeared to have limited opportunity for divine action. The theological response to this world picture was deism: God created the world but it runs on its own. This world picture collapsed in the early twentieth century as quantum mechanics was discovered and developed into a central, if not the central, part of scientific understanding of physical reality. The quantum-mechanical world picture cannot be modelled after the gears, pulleys, and levers of the classical world: its irreducible constituent parts are more like tiny clouds than tiny gears. Furthermore, the behaviour of these smallest parts was discovered to be indeterministic, and eternally so until observed by a mind. The mysterious behaviour of quantum particles, which compose all reality, and the puzzling role of minds invited speculation that God may be acting through the natural order to influence events. Such divine action could exist entirely within the envelope of possibilities possessed by quantum particles and thus not require any disruption of the natural order.

𝒫-characters and the structure of finite solvable groups

Let 𝐺 be a finite group. An irreducible character of 𝐺 is called a 𝒫-character if it is an irreducible constituent of (1_{H})^{G} for some maximal subgroup 𝐻 of 𝐺. In this paper, we obtain some conditions for a solvable group 𝐺 to be 𝑝-nilpotent or 𝑝-closed in terms of 𝒫-characters.

Criticality as a Determinant of Integrated Information Φ in Human Brain Networks

Integrated information theory (IIT) describes consciousness as information integrated across highly differentiated but irreducible constituent parts in a system. However, in a complex dynamic system such as the brain, the optimal conditions for large integrated information systems have not been elucidated. In this study, we hypothesized that network criticality, a balanced state between a large variation in functional network configuration and a large constraint on structural network configuration, may be the basis of the emergence of a large Φ, a surrogate of integrated information. We also hypothesized that as consciousness diminishes, the brain loses network criticality and Φ decreases. We tested these hypotheses with a large-scale brain network model and high-density electroencephalography (EEG) acquired during various levels of human consciousness under general anesthesia. In the modeling study, maximal criticality coincided with maximal Φ. The EEG study demonstrated an explicit relationship between Φ, criticality, and level of consciousness. The conscious resting state showed the largest Φ and criticality, whereas the balance between variation and constraint in the brain network broke down as the response rate dwindled. The results suggest network criticality as a necessary condition of a large Φ in the human brain.

Multiplication of Polynomials on Hermitian Symmetric spaces and Littlewood–Richardson Coefficients

Let K be a complex reductive algebraic group and V a representation of K. Let S denote the ring of polynomials on V. Assume that the action of K on S is multiplicity-free. If ƛ denotes the isomorphism class of an irreducible representation of K, let ρƛ : K → GL(Vƛ) denote the corresponding irreducible representation and Sƛ the ƛ-isotypic component of S. Write Sƛ ・ Sμ for the subspace of S spanned by products of Sƛ and Sμ. If Vν occurs as an irreducible constituent of Vƛ ⊗ Vμ, is it true that Sν ⊆ Sƛ ・ Sμ? In this paper, the authors investigate this question for representations arising in the context of Hermitian symmetric pairs. It is shown that the answer is yes in some cases and, using an earlier result of Ruitenburg, that in the remaining classical cases, the answer is yes provided that a conjecture of Stanley on the multiplication of Jack polynomials is true. It is also shown how the conjecture connects multiplication in the ring S to the usual Littlewood–Richardson rule.

Nonabelian Fully-Ramified Sections

Let G be a finite group and let K and L be normal subgroups of G such that |K : L| and |G : K| are relatively prime, and assume that |K : L| is odd. Let H be a subgroup of G such that G = HK and H ∩ K = L. Let φ be an irreducible character of L that is invariant under the action of L and is fully ramified with respect to K/L. If χ ∈ Irr(G) is a constituent of φG, then we prove that χH has a unique irreducible constituent having odd multiplicity.

On tensor induction of group representations

Let G be a (not necessarily finite) group and ρ a finite dimensional faithful irreducible representation of G over an arbitrary field; write ρ¯ for ρ viewed as a projective representation. Suppose that ρ is not induced (from any proper subgroup) and that ρ¯ is not a tensor product (of projective representations of dimension greater than 1). Let K be a noncentral subgroup which centralizes all its conjugates in G except perhaps itself, write H for the normalizer of K in G, and suppose that some irreducible constituent, σ say, of the restriction p↓K is absolutely irreducible. It is proved that then (ρ is absolutely irreducible and) ρ¯ is tensor induced from a projective representation of H, namely from a tensor factor π of ρ¯↓H such that π↓K = σ¯ and ker π is the centralizer of K in G.