The permutation action of finite symplectic groups of odd characteristic on their standard modules

Motivated by the incidence problems between points and flats of a symplectic polar space, we study a large class of submodules of the space of functions on the standard module of a finite symplectic group of odd characteristic. Our structure results on this class of submodules allow us to determine the $p$-ranks of the incidence matrices between points and flats of the symplectic polar space. In particular, we give an explicit formula for the $p$-rank of the generalized quadrangle ${\rm W}(3,q)$, where $q$ is an odd prime power. Combined with the earlier results of Sastry and Sin on the 2-rank of ${\rm W}(3,2^t)$, it completes the determination of the $p$-ranks of ${\rm W}(3,q)$.Comment: 22 page

Evolutionary outcomes for pairs of planets undergoing orbital migration and circularization: second order resonances and observed period ratios in Kepler's planetary systems

In order to study the origin of the architectures of low mass planetary systems, we perform numerical surveys of the evolution of pairs of coplanar planets in the mass range (1-4)\ \rmn{M}_{\oplus}. These evolve for up to 2\times10^7 \rmn{yr} under a range of orbital migration torques and circularization rates assumed to arise through interaction with a protoplanetary disc. Near the inner disc boundary, significant variations of viscosity, interaction with density waves or with the stellar magnetic field could occur and halt migration, but allow ircularization to continue. This was modelled by modifying the migration and circularization rates. Runs terminated without an extended period of circularization in the absence of migration torques gave rise to either a collision, or a system close to a resonance. These were mostly first order with a few $\%$ terminating in second order resonances. Both planetary eccentricities were small $< 0.1$ and all resonant angles liberated. This type of survey produced only a limited range of period ratios and cannot reproduce Kepler observations. When circularization alone operates in the final stages, divergent migration occurs causing period ratios to increase. Depending on its strength the whole period ratio range between $1$ and $2$ can be obtained. A few systems close to second order commensurabilities also occur. In contrast to when arising through convergent migration, resonant trapping does not occur and resonant angles circulate. Thus the behaviour of the resonant angles may indicate the form of migration that led to near resonance.Comment: 15 pages, 12 figures, 2014, MNRAS, 449, 304