Toric degenerations of Grassmannians and Schubert varieties from matching field tableaux

We study the combinatorics of Gr\"obner degenerations of Grassmannians and the Schubert varieties inside them. We provide a family of binomial ideals whose combinatorics is governed by tableaux induced by matching fields in the sense of Sturmfels and Zelevinsky. We prove that these ideals are all quadratically generated and they yield a SAGBI basis of the Pl\"ucker algebra. This leads to a new family of toric degenerations of Grassmannians. Moreover, we apply our results to construct a family of Gr\"obner degenerations of Schubert varieties inside Grassmannians. We provide a complete characterization of toric ideals among these degenerations in terms of the combinatorics of matching fields, permutations, and semi-standard tableaux

Toric degenerations of flag varieties from matching field tableaux

We present families of tableaux which interpolate between the classical semi-standard Young tableaux and matching field tableaux. Algebraically, this corresponds to SAGBI bases of Pl\"ucker algebras. We show that each such family of tableaux leads to a toric ideal, that can be realized as initial of the Pl\"ucker ideal, hence a toric degeneration for the flag variety

Standard monomial theory and toric degenerations of Schubert varieties from matching field tableaux

We study Gr\"obner degenerations of Schubert varieties inside flag varieties. We consider toric degenerations of flag varieties induced by matching fields and semi-standard Young tableaux. We describe an analogue of matching field ideals for Schubert varieties inside the flag variety and give a complete characterization of toric ideals among them. We use a combinatorial approach to standard monomial theory to show that block diagonal matching fields give rise to toric degenerations. Our methods and results use the combinatorics of permutations associated to Schubert varieties, matching fields and their corresponding tableaux

The expression of stlA in Photorhabdus luminescens is controlled by nutrient limitation

Photorhabdus is a genus of Gram-negative entomopathogenic bacteria that also maintain a mutualistic association with nematodes from the family Heterorhabditis. Photorhabdus has an extensive secondary metabolism that is required for the interaction between the bacteria and the nematode. A major component of this secondary metabolism is a stilbene molecule, called ST. The first step in ST biosynthesis is the non-oxidative deamination of phenylalanine resulting in the production of cinnamic acid. This reaction is catalyzed by phenylalanine-ammonium lyase, an enzyme encoded by the stlA gene. In this study we show, using a stlA-gfp transcriptional fusion, that the expression of stlA is regulated by nutrient limitation through a regulatory network that involves at least 3 regulators. We show that TyrR, a LysR-type transcriptional regulator that regulates gene expression in response to aromatic amino acids in E. coli, is absolutely required for stlA expression. We also show that stlA expression is modulated by σS and Lrp, regulators that are implicated in the regulation of the response to nutrient limitation in other bacteria. This work is the first that describes pathway-specific regulation of secondary metabolism in Photorhabdus and, therefore, our study provides an initial insight into the complex regulatory network that controls secondary metabolism, and therefore mutualism, in this model organism

Conditional independence ideals with hidden variables

We study a class of determinantal ideals that are related to conditional independence (CI) statements with hidden variables. Such CI statements correspond to determinantal conditions on a matrix whose entries are probabilities of events involving the observed random variables. We focus on an example that generalizes the CI ideals of the intersection axiom. In this example, the minimal primes are again determinantal ideals, which is not true in general.Comment: 20 pages, 1 figure, 4 table

The equivariant Ehrhart theory of polytopes with order-two symmetries

We study the equivariant Ehrhart theory of families of polytopes that are invariant under a non-trivial action of the group with order two. We study families of polytopes whose equivariant $H^*$-polynomial both succeed and fail to be effective, in particular, the symmetric edge polytopes of cycles and the rational cross-polytope. The latter provides a counterexample to the effectiveness conjecture if the requirement that the vertices of the polytope have integral coordinates is loosened to allow rational coordinates. Moreover, we exhibit such a counterexample whose Ehrhart function has period one and coincides with the Ehrhart function of a lattice polytope.Comment: 16 page

A note on the Cops & Robber game on graphs embedded in non-orientable surfaces

The Cops and Robber game is played on undirected finite graphs. A number of cops and one robber are positioned on vertices and take turns in sliding along edges. The cops win if they can catch the robber. The minimum number of cops needed to win on a graph is called its cop number. It is known that the cop number of a graph embedded on a surface $X$ of genus $g$ is at most $3g/2 + 3$, if $X$ is orientable (Schroeder 2004), and at most $2g+1$, otherwise (Nowakowski & Schroeder 1997). We improve the bounds for non-orientable surfaces by reduction to the orientable case using covering spaces. As corollaries, using Schroeder's results, we obtain the following: the maximum cop number of graphs embeddable in the projective plane is 3; the cop number of graphs embeddable in the Klein Bottle is at most 4, and an upper bound is $3g/2 + 3/2$ for all other $g$.Comment: 5 pages, 1 figur

Bifunctional catalytic electrode

The present invention relates to an oxygen electrode for a unitized regenerative hydrogen-oxygen fuel cell and the unitized regenerative fuel cell having the oxygen electrode. The oxygen electrode contains components electrocatalytically active for the evolution of oxygen from water and the reduction of oxygen to water, and has a structure that supports the flow of both water and gases between the catalytically active surface and a flow field or electrode chamber for bulk flow of the fluids. The electrode has an electrocatalyst layer and a diffusion backing layer interspersed with hydrophilic and hydrophobic regions. The diffusion backing layer consists of a metal core having gas diffusion structures bonded to the metal core

Conditional probabilities via line arrangements and point configurations

We study the connection between probability distributions satisfying certain conditional independence (CI) constraints, and point and line arrangements in incidence geometry. To a family of CI statements, we associate a polynomial ideal whose algebraic invariants are encoded in a hypergraph. The primary decompositions of these ideals give a characterisation of the distributions satisfying the original CI statements. Classically, these ideals are generated by 2-minors of a matrix of variables, however, in the presence of hidden variables, they contain higher degree minors. This leads to the study of the structure of determinantal hypergraph ideals whose decompositions can be understood in terms of point and line configurations in the projective space.Comment: 24 pages, 5 figure