Related papers: On decomposing Betti tables and $O$-sequences

We give a decomposition as a direct sum of indecomposable modules of several types of Specht modules in characteristic $2$. These include the Specht modules labelled by hooks, whose decomposability was considered by Murphy. Since the main…

The modular decomposition of a graph $G$ is a natural construction to capture key features of $G$ in terms of a labeled tree $(T,t)$ whose vertices are labeled as "series" ($1$), "parallel" ($0$) or "prime". However, full information of $G$…

Let $G$ be a finite simple graph on the vertex set $V(G)$ and let $r$ be a positive integer. We consider the hypergraph $\mathrm{Con}_r(G)$ whose vertices are the vertices of $G$ and the (hyper)edges are all $A\subseteq V(G)$ such that…

Let $I$ be a perfect ideal of height 3 in a Gorenstein local ring $R$. Let $\mathbb{F}$ be the minimal free resolution of $I$. A sequence of linear maps, which generalize the multiplicative structure of $\mathbb{F}$, can be defined using…

We associate a sequence of positive integers, termed the type sequence, with a cochordal graph. Using this type sequence, we compute all graded Betti numbers of its edge ideal. We then classify all positive integer $n$ such that the zero…

We give a sufficient condition for a monomial ideal to have a nonzero Betti number in each multidegree. In the case of facet ideals of simplicial forests, this condition becomes a necessary one and it allows us to characterize Betti…

We introduce the notion of Betti category for graded modules over suitably graded polynomial rings, and more generally for modules over certain small categories. Our categorical approach allows us to treat simultaneously many important…

Sharply o-minimal structures (denoted \so-minimal) are a strict subclass of the o-minimal structures, aimed at capturing some finer features of structures arising from algebraic geometry and Hodge theory. Sharp o-minimality associates to…

Suppose $S$ is an affine, noetherian scheme, $X$ is a separated, noetherian $S$-scheme, $\mathcal{E}$ is a coherent ${\mathcal{O}}_{X}$-bimodule and $\mathcal{I} \subset T(\mathcal{E})$ is a graded ideal. We study the geometry of the…

Boij and S\"oderberg made a pair of conjectures, which were subsequently proven by Eisenbud and Schreyer and then extended by Boij and S\"oderberg, about the structure of Betti diagrams of Graded modules. In the theory, a particular family…

A Betti splitting $I=J+K$ of a monomial ideal $I$ ensures the recovery of the graded Betti numbers of $I$ starting from those of $J,K$ and $J \cap K$. In this paper, we introduce this condition for simplicial complexes, and, by using…

The concept of descent algebras over a field of characteristic zero is extended to define descent algebras over a field of prime characteristic. Some basic algebraic structure of the latter, including its radical and irreducible modules, is…

An explicit construction is given of a minimal free resolution of the ideal generated by all squarefree monomials of a given degree. The construction relies upon and exhibits the natural action of the symmetric group on the syzygy modules.…

Let $G$ be a finite simple graph on $n$ non-isolated vertices, and let $J_G$ be its binomial edge ideal. We determine almost all pairs $(\text{projdim}(J_G),\text{reg}(J_G))$, where $G$ ranges over all finite simple graphs on $n$…

This paper studies a class of binomial ideals associated to graphs with finite vertex sets. They generalize the binomial edge ideals, and they arise in the study of conditional independence ideals. A Gr\"obner basis can be computed by…

We describe a recursive algorithm that decomposes an algebraic set into locally closed equidimensional sets, i.e. sets which each have irreducible components of the same dimension. At the core of this algorithm, we combine ideas from the…

A decomposition space (also called 2-Segal space) is a simplicial object satisfying an exactness condition weaker than the Segal condition: just as the Segal condition expresses composition, the new condition expresses decomposition. It is…

The bigraded Betti numbers b^{-i,2j}(P) of a simple polytope P are the dimensions of the bigraded components of the Tor groups of the face ring k[P]. The numbers b^{-i,2j}(P) reflect the combinatorial structure of P as well as the topology…

Let $o$ be the ring of integers in a finite extension $K/\mathbb{Q}_p$ and $G=\mathbf{G}(\mathbb{Q}_p)$ be the $\mathbb{Q}_p$-points of a $\mathbb{Q}_p$-split reductive group $\mathbf{G}$ defined over $\mathbb{Z}_p$ with connected centre…

Let $\mathfrak{g}$ be a finite-dimensional simple Lie algebra over an algebraically closed field of characteristic 0. In this paper we classify all regular decompositions of $\mathfrak{g}$ and its irreducible root system $\Delta$. A regular…