On dimension growth of modular irreducible representations of semisimple Lie algebras

In this paper we investigate the growth with respect to $p$ of dimensions of irreducible representations of a semisimple Lie algebra $\mathfrak{g}$ over $\overline{\mathbb{F}}_p$. More precisely, it is known that for $p\gg 0$, the irreducibles with a regular rational central character $\lambda$ and $p$-character $\chi$ are indexed by a certain canonical basis in the $K_0$ of the Springer fiber of $\chi$. This basis is independent of $p$. For a basis element, the dimension of the corresponding module is a polynomial in $p$. We show that the canonical basis is compatible with the two-sided cell filtration for a parabolic subgroup in the affine Weyl group defined by $\lambda$. We also explain how to read the degree of the dimension polynomial from a filtration component of the basis element. We use these results to establish conjectures of the second author and Ostrik on a classification of the finite dimensional irreducible representations of W-algebras, as well as a strengthening of a result by the first author with Anno and Mirkovic on real variations of stabilities for the derived category of the Springer resolution.Comment: 24 pages, v2 acknowledgements added; v3 references adde

Correlations of Different Surface Tests: Tire Behavior Math Model for the High Speed Civil Transport (HSCT) and Michelin Tire Properties Tests for Boeing 777

In the surfaces correlation study, several different volumetric and drainage measurement techniques for classifying surface texture were evaluated as part of a major study to develop and improve methods for predicting tire friction performance on all types of pavement. The objective of the evaluation was to seek relationships between the different techniques, and to relate those results to surface frictional characteristics. We needed to know how each of the tests could be related to each other. Another of my assigned projects was to make a tire behavior math model for the High Speed Civil Transport (HSCT) using the same methods used for the space shuttle a few years ago. A provided third order equation with two variables was used. This model will also be used for studies with the Boeing 777. Only a few changes will be necessary to adapt it for this other aircraft, which is the newest offered by Boeing. In my final project I was involved with testing the tires for this new aircraft using the Aircraft Landing Dynamics Facility (ALDF) test carriage within the carriage house at LaRC. A 50 inch diameter radial tire manufactured by Michelin Aircraft Tire Corporation had to be tested to double overload of 114,000 pounds. The rated load of each tire is 57,000 pounds, but Boeing required tests assuming failure of a companion tire that could have cost Michelin approximately $12 million to build a facility to provide the required test capability. Here at LaRC, only minimum modifications to the facility were required to perform this specific test