Initial ideals of tangent cones to Schubert varieties in orthogonal Grassmannians

We compute the initial ideals, with respect to certain conveniently chosen term orders, of ideals of tangent cones at torus fixed points to Schubert varieties in orthogonal Grassmannians. The initial ideals turn out to be square-free monomial ideals and therefore Stanley-Reisner face rings of simplicial complexes. We describe these complexes. The maximal faces of these complexes encode certain sets of non-intersecting lattice paths.Comment: 36 pages, 3 figures; for version 2: error in definition of "new form" correcte

The cell lineage of the muscles of the Drosophila head

Using a cell marker mutation the cell lineage of the muscles of the Drosophila head are traced out. Three sets of muscles separated by lineage restrictions are observed, even when cells are marked as early as the blastoderm stage. Each set underlies the derivatives of one of the three pairs of imaginal discs which differentiate to form the epidermis of the adult head. Clones of the homoeotic mutation engrailed (en^10) were apparently normal in the muscles of the head. The muscle clone frequency, at the blastoderm stage, in each hemisegment of the fly is similar, indicating an equal partitioning of cells during segmentation

Stability of the Chari-Pressley-Loktev bases for local Weyl modules of sl2[t]sl_2[t]

We prove stability of the Chari-Pressley-Loktev bases for natural inclusions of local Weyl modules of the current algebra $sl_2[t]$. These modules being known to be Demazure submodules in the level 1 representations of the affine Lie algebra $\widehat{sl_2}$, we obtain, by passage to the direct limit, bases for the level 1 representations themselves.Comment: 20 pages; minor revision