Extensions, Levi subgroups and character formulas

This paper consists of three interconnected parts. Parts I,III study the relationship between the cohomology of a reductive group and that of a Levi subgroup. For example, we provide a necessary condition, arising from Kazhdan-Lusztig theory, for the natural map on Ext-groups of irreducible modules to be surjective. In cohomological degree 1, the map is always an isomorphism, under our hypothesis. These results were inspired by recent work of Hemmer obtained for general linear groups, and they both extend and improve upon his work when our condition is met. Part II obtains results on Lusztig character formulas for reductive groups, obtaining new necessary and sufficient conditions for such formulas to hold. In the special case of general linear groups, these conditions can be recast in a striking way completely in terms of explicit representation theoretic properties of the symmetric group (and the results improve upon the sufficient cohomological conditions established recently in the authors)

On pp-filtrations of Weyl modules

This paper considers Weyl modules for a simple, simply connected algebraic group over an algebraically closed field $k$ of positive characteristic $p\not=2$. The main result proves, if $p\geq 2h-2$ (where $h$ is the Coxeter number) and if the Lusztig character formula holds for all (irreducible modules with) regular restricted highest weights, then any Weyl module $\Delta(\lambda)$ has a $\Delta^p$-filtration, namely, a filtration with sections of the form $\Delta^p(\mu_0+p\mu_1):=L(\mu_0)\otimes\Delta(\mu_1)^{[1]}$, where $\mu_0$ is restricted and $\mu_1$ is arbitrary dominant. In case the highest weight $\lambda$ of the Weyl module $\Delta(\lambda)$ is $p$-regular, the $p$-filtration is compatible with the $G_1$-radical series of the module. The problem of showing that Weyl modules have $\Delta^p$-filtrations was first proposed as a worthwhile ("w\"unschenswert") problem in Jantzen's 1980 Crelle paper.Comment: Latest version corrects minor mistakes in previous versions. A reference is made to Williamson's recent arXiv posting, providing some relevant discussion in a footnote. [Comments on earlier versions: Previous v. 1 with minor errors and statements corrected. Improved organization. Should replace v. 2 which is an older version (even older than v.1) and was mistakenly posted.

Parshall flumes of large size

CER53RLP9.March 1953.United States Department of Agriculture, Soil Conservation Service, Division of Irrigation Engineering and Water Conservation in cooperation with Colorado Agricultural Experiment Station.Revision of Exp. Sta. Bul. 386.Revised edition prepared by Carl Rohwer under the direction of George D. Clyde

Model and prototype studies for the design of sand traps

CER47-52RLP36.Presented at the Summer Convention, American Society of Civil Engineers, Irrigation Division, Toronto, Canada, July 14, 1950