Maximal compatible splitting and diagonals of Kempf varieties

Lakshmibai, Mehta and Parameswaran (LMP) introduced the notion of maximal multiplicity vanishing in Frobenius splitting. In this paper we define the algebraic analogue of this concept and construct a Frobenius splitting vanishing with maximal multiplicity on the diagonal of the full flag variety. Our splitting induces a diagonal Frobenius splitting of maximal multiplicity for a special class of smooth Schubert varieties first considered by Kempf. Consequences are Frobenius splitting of tangent bundles, of blow-ups along the diagonal in flag varieties along with the LMP and Wahl conjectures in positive characteristic for the special linear group.Comment: Revised according to referee suggestions. To appear in Annales de l'Institut Fourie

A conjectural generalization of n! result to arbitrary groups

We relate the n! conjecture (by Garsia and Haiman) to the geometry of principal nilpotent pairs, and state a conjecture generalizing the n! conjecture to arbitrary semisimple algebraic groups. We also show, using Borel's fixed point theorem, how to reduce the n! conjecture to staircase partitions. Finally we study the interplay between characteristic p and the n! conjecture for box partitions.Comment: Main conjectures has changed, 28 page

Rotiferan Hox genes give new insights into the evolution of metazoan bodyplans

The phylum Rotifera consists of minuscule, nonsegmented animals with a unique body plan and an unresolved phylogenetic position. The presence of pharyngeal articulated jaws supports an inclusion in Gnathifera nested in the Spiralia. Comparison of Hox genes, involved in animal body plan patterning, can be used to infer phylogenetic relationships. Here, we report the expression of five Hox genes during embryogenesis of the rotifer Brachionus manjavacas and show how these genes define different functional components of the nervous system and not the usual bilaterian staggered expression along the anteroposterior axis. Sequence analysis revealed that the lox5-parapeptide, a key signature in lophotrochozoan and platyhelminthean Hox6/lox5 genes, is absent and replaced by different signatures in Rotifera and Chaetognatha, and that the MedPost gene, until now unique to Chaetognatha, is also present in rotifers. Collectively, our results support an inclusion of chaetognaths in gnathiferans and Gnathifera as sister group to the remaining spiralians

Closures of Steinberg fibers in twisted wonderful compactifications

By a case-free approach we give a precise description of the closure of a Steinberg fiber within a twisted wonderful compactification of a simple linear algebraic group. In the non-twisted case this description was earlier obtained by the first author.Comment: 12 pages, final versio