Sheaves on the alcoves and modular representations II

We relate the category of sheaves on alcoves that was constructed in "Sheaves on the alcoves and modular representations I" to the representation theory of reductive algebraic groups. In particular, we show that its indecomposable projective objects encode the simple rational characters of a reductive algebraic group in all characteristics above the Coxeter number.Comment: 38 pages. Some corrections in version 3. This is the companion article to "Sheaves on the alcoves and modular representations I". Both articles replace our earlier submissions arXiv:1508.05579 and arXiv:1504.0169

Following Schubert varieties under Feigin's degeneration of the flag variety

We describe the effect of Feigin's flat degeneration of the type $\textrm{A}$ flag variety on its Schubert varieties. In particular, we study when they stay irreducible and in several cases we are able to encode reducibility of the degenerations in terms of symmetric group combinatorics. As a side result, we obtain an identification of some Schubert varieties with Richardson varieties in higher rank partial flag varieties.Comment: Comments are welcome! 22 pages + Appendix (5 pages), 2 figure

Degenerate flag varieties and Schubert varieties: a characteristic free approach

We consider the PBW filtrations over the integers of the irreducible highest weight modules in type A and C. We show that the associated graded modules can be realized as Demazure modules for group schemes of the same type and doubled rank. We deduce that the corresponding degenerate flag varieties are isomorphic to Schubert varieties in any characteristic.Comment: 23 pages; A few typos corrected; Authors affiliation adde

Development of an Ex Vivo Organ Culture Technique to Evaluate Probiotic Utilization in IBD

The consistent technical and conceptual progress in the study of the microbiota has led novel impulse to the research for therapeutical application of probiotic bacteria in human pathologies, such as inflammatory bowel disease (IBD). Considering the heterogenous results of probiotics in clinical studies, the model of translational medicine may lead to a more specific and efficacious utilization of probiotic bacteria in IBD. In this regard, the selection and utilization of appropriate experimental models may drive the transition from pure in vitro systems to practical clinical application. We developed a simple and reproducible ex vivo organ culture method with potential utilization for the evaluation of probiotic bacteria efficacy in IBD patients