Rank Bounded Hibi Subrings for Planar Distributive Lattices

Let $L$ be a distributive lattice and $R[L]$ the associated Hibi ring. We show that if $L$ is planar, then any bounded Hibi subring of $R[L]$ has a quadratic Gr\"obner basis. We characterize all planar distributive lattices $L$ for which any proper rank bounded Hibi subring of $R[L]$ has a linear resolution. Moreover, if $R[L]$ is linearly related for a lattice $L$, we find all the rank bounded Hibi subrings of $R[L]$ which are linearly related too.Comment: Accepted in Mathematical Communication

Certain Classes of Cohen-Macaulay Multipartite Graphs

The Cohen-Macaulay property of a graph arising from a poset has been studied by various authors. In this article, we study the Cohen-Macaulay property of a graph arising from a family of reflexive and antisymmetric relations on a set. We use this result to find classes of multipartite graphs which are Cohen-Macaulay

Rank bounded Hibi subrings for planar distributive lattices

Let L be a distributive lattice and R[L] the associated Hibi ring. We show that if L is planar, then any bounded Hibi subring of R[L] has a quadratic Grobner basis. We characterize all planar distributive lattices L for which any proper rank bounded Hibi subring of R[L] has a linear resolution. Moreover, if R[L] is linearly related for a lattice L, we find all the rank bounded Hibi subrings of R[L] which are linearly related too

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

Resolutions of letterplace ideals of posets

We investigate resolutions of letterplace ideals of posets. We develop topological results to compute their multigraded Betti numbers, and to give structural results on these Betti numbers. If the poset is a union of no more than $c$ chains, we show that the Betti numbers may be computed from simplicial complexes of no more than $c$ vertices. We also give a recursive procedure to compute the Betti diagrams when the Hasse diagram of $P$ has tree structure.Comment: 21 page