Related papers: Gr\"obner geometry for regular nilpotent Hessenber…

In this paper, we study the subvarieties of a complex flag variety that are invariant under the action of a maximal torus. Using combinatorial techniques derived from matroid theory, we introduce a decomposition of this variety into affine,…

We compute the reduced Gr\"{o}bner basis of the toric ideal with respect to a suitable monomial order and we study the Hilbert series of the vertex cover algebra $A(G)$, where $G$ is an unmixed bipartite graph without isolated vertices.

We describe a Schubert induction theorem, a tool for analyzing intersections on a Grassmannian over an arbitrary base ring. The key ingredient in the proof is the Geometric Littlewood-Richardson rule, described in a companion paper.…

Let $\mathbf{G}$ be one of the ind-groups $GL(\infty)$, $O(\infty)$, $Sp(\infty)$ and $\mathbf{P}\subset \mathbf{G}$ be a splitting parabolic ind-subgroup. The ind-variety $\mathbf{G}/\mathbf{P}$ has been identified with an ind-variety of…

We revisit tensor algebras of subproduct systems with Hilbert space fibers, resolving some open questions in the case of infinite dimensional fibers. We characterize when a tensor algebra can be identified as the algebra of uniformly…

We study an inductive method of computing initial ideals and Gr\"obner bases for families of ideals in a polynomial ring. This method starts from a given set of pairs $(I,J)$ where $I$ is any ideal and $J$ is a monomial ideal contained in…

It is shown that for a normal subgroup $N$ of a group $G$, $G/N$ cyclic, the kernel of the map $N^{\mathrm{ab}}\to G^{\mathrm{ab}}$ satisfies the classical Hilbert 90 property (cf. Thm. A). As a consequence, if $G$ is finitely generated,…

The nonsingular variety G_T parametrizes all graded ideals I of R=k[x,y] for which the Hilbert function H(R/I)=T. The variety G_T has a natural cellular decomposition: each cell V(E) corresponds to a monomial ideal E for which H(R/E)=T.…

The MinRank problem is a simple linear algebra problem: given matrices with coefficients in a field, find a non trivial linear combination of the matrices that has a small rank. There are several algebraic modeling of the problem. The main…

We study lowest-weight irreducible representations of rational Cherednik algebras attached to the complex reflection groups G(m,r,n) in characteristic p. Our approach is mostly from the perspective of commutative algebra. By studying the…

Let $G$ be a finite abelian group and let $K$ be an algebraically closed field of characteristic 0. We consider associative unital algebras $A$ over $K$ graded by $G$, that is $A=\oplus_{g\in G} A_g$, where the vector subspaces $A_g$…

Let $R$ be a finite commutative ring with identity, and let $P$ be a proper prime ideal of $R$. The prime ideal graph $\Gamma_P(R)$ has vertex set of $R\setminus\{0\}$, where two distinct vertices $x$ and $y$ are adjacent if and only if…

We propose new structures called almost o-minimal structures and $\mathfrak X$-structures. The former is a first-order expansion of a dense linear order without endpoints such that the intersection of a definable set with a bounded open…

Given a semisimple complex linear algebraic group $G$ and a lower ideal $I$ in positive roots of $G$, three objects arise: the ideal arrangement $\mathcal{A}_I$, the regular nilpotent Hessenberg variety $\mbox{Hess}(N,I)$, and the regular…

For any complex reductive connected Lie group G, many of the structure constants of the ordinary cohomology ring H^*(G/B; Z) vanish in the Schubert basis, and the rest are strictly positive. We present a combinatorial game, the ``root…

A Hilbert basis is a set of vectors X such that the integer cone (semigroup) generated by X is the intersection of the lattice generated by X with the cone generated by X. Define a graph to be (cut) Hilbert if its set of cuts forms a…

In this paper, a description of the set-theoretical defining equations of symplectic (type C) Grassmannian/flag/Schubert varieties in corresponding (type A) algebraic varieties is given as linear polynomials in Pl$\ddot{u}$cker coordinates,…

We generalize some properties related to Hilbert series and Lefschetz properties of Milnor algebras of projective hypersurfaces with isolated singularities to the more general case of an almost complete intersection ideal $J$ of dimension…

Let $\Omega \subseteq \mathbb C^m$ be a bounded connected open set and $\mathcal H \subseteq \mathcal O(\Omega)$ be an analytic Hilbert module, i.e., the Hilbert space $\mathcal H$ possesses a reproducing kernel $K$, the polynomial ring…

Schubert varieties have been exhaustively studied with a plethora of techniques: Coxeter groups, explicit desingularization, Frobenius splitting, etc. Many authors have applied these techniques to various other varieties, usually defined by…