arXiv subjects

Mike Cummings

Publications and source records attributed to Mike Cummings.

4 recordsLinked to original sources

arXiv2026

Webs and smooth components of two column Springer fibers

Webs and Springer fibers are separately important objects in representation theory: webs give a diagrammatic calculus for tensor invariants of $\mathfrak{sl}_k$, and the cohomology group of Springer fibers can be used to construct the irreducible representations of the symmetric group. Fung's 1997 thesis gave the first evidence of a connection between $\mathfrak{sl}_2$ webs and Springer fibers, showing that webs naturally index and describe the components of certain "two row" Springer fibers. However, this case is known to be far from generic. This paper deepens this connection with a similar correspondence in the substantially more complicated "two column" case. In particular, and building on works of Fresse, Melnikov, and Sakas-Obeid, we use webs to give a clean characterization of the smooth components of two column rectangle Springer fibers and a simple description of the geometry of these smooth components. We also show that the Poincar\'e polynomial of the smooth components is invariant under the natural dihedral action on the corresponding webs.

math.AG↗

arXiv2023

Gr\"obner geometry for regular nilpotent Hessenberg Schubert cells

A regular nilpotent Hessenberg Schubert cell is the intersection of a regular nilpotent Hessenberg variety with a Schubert cell. In this paper, we describe a set of minimal generators of the defining ideal of a regular nilpotent Hessenberg Schubert cell in the type $A$ setting. We show that these minimal generators are a Gr\"obner basis for an appropriate lexicographic monomial order. As a consequence, we obtain a new computational-algebraic proof, in type $A$, of Tymoczko's result that regular nilpotent Hessenberg varieties are paved by affine spaces. In addition, we prove that these defining ideals are complete intersections, are geometrically vertex decomposable, and compute their Hilbert series. We also produce a Frobenius splitting of each Schubert cell that compatibly splits all of the regular nilpotent Hessenberg Schubert cells contained in it. This work builds on, and extends, work of the second and third author on defining ideals of intersections of regular nilpotent Hessenberg varieties with the (open) Schubert cell associated to the Bruhat-longest permutation.

math.AG↗

arXiv2022

The GeometricDecomposability package for Macaulay2

Using the geometric vertex decomposition property first defined by Knutson, Miller, and Yong, a recursive definition for geometrically vertex decomposable ideals was given by Klein and Rajchgot. We introduce the Macaulay2 package GeometricDecomposability which provides a suite of tools to experiment and test the geometric vertex decomposability property of an ideal.

math.AC↗

arXiv2022

Geometric vertex decomposition and liaison for toric ideals of graphs

The geometric vertex decomposability property for polynomial ideals is an ideal-theoretic generalization of the vertex decomposability property for simplicial complexes. Indeed, a homogeneous geometrically vertex decomposable ideal is radical and Cohen-Macaulay, and is in the Gorenstein liaison class of a complete intersection (glicci). In this paper, we initiate an investigation into when the toric ideal $I_G$ of a finite simple graph $G$ is geometrically vertex decomposable. We first show how geometric vertex decomposability behaves under tensor products, which allows us to restrict to connected graphs. We then describe a graph operation that preserves geometric vertex decomposability, thus allowing us to build many graphs whose corresponding toric ideals are geometrically vertex decomposable. Using work of Constantinescu and Gorla, we prove that toric ideals of bipartite graphs are geometrically vertex decomposable. We also propose a conjecture that all toric ideals of graphs with a square-free degeneration with respect to a lexicographic order are geometrically vertex decomposable. As evidence, we prove the conjecture in the case that the universal Gr\"obner basis of $I_G$ is a set of quadratic binomials. We also prove that some other families of graphs have the property that $I_G$ is glicci.

math.AC↗