About

Patricia Klein is a professor in the Department of Horticultural Sciences at Texas A&M University. She holds a Bachelor of Science degree in Horticulture from Texas A&M University and a Ph.D. in Biochemistry from the same institution. Her academic and professional background is rooted in horticulture and biochemistry, and she is a member of Texas A&M AgriLife, which encompasses Texas A&M AgriLife Extension Service, Texas A&M AgriLife Research, Texas A&M Forest Service, Texas A&M Veterinary Medical Diagnostic Lab, and the College of Agriculture & Life Sciences. Her office is located in HFSB 416 at Texas A&M University, College Station, TX. Further details about her specific research focus or key contributions are not provided on the page.

Research topics

• Pure mathematics
• Computer Science
• Mathematics
• Artificial Intelligence
• Programming language
• Chemistry
• Discrete mathematics
• Physics
• Mathematical analysis

Selected publications

• Simplicial Complexes and Matroids with Vanishing $T^2$

  The Electronic Journal of Combinatorics · 2025-04-24

  articleOpen access

  We investigate quotients by radical monomial ideals for which $T^2$, the second cotangent cohomology module, vanishes. The dimension of the graded components of $T^2$, and thus their vanishing, depends only on the combinatorics of the corresponding simplicial complex. We give both a complete characterization and a full list of one dimensional complexes with $T^ 2 = 0$. We characterize the graded components of $T^ 2$ when the simplicial complex is a uniform matroid. Finally, we show that $T^2$ vanishes for all matroids of corank at most two and conjecture that all connected matroids with vanishing $T^2$ are of corank at most two.

  PublisherOA PDFDOI

• Some algebraic properties of ASM varieties

  arXiv (Cornell University) · 2025-05-15

  preprintOpen access

  Fulton's matrix Schubert varieties are affine varieties that arise in the study of Schubert calculus in the complete flag variety. Weigandt showed that arbitrary intersections of matrix Schubert varieties, now called ASM varieties, are indexed by alternating sign matrices (ASMs), objects with a long history in enumerative combinatorics. It is very difficult to assess Cohen-Macaulayness of ASM varieties or to compute their codimension, though these properties are well understood for matrix Schubert varieties due to work of Fulton. In this paper we study these properties of ASM varieties with a focus on the relationship between a pair of ASMs and their direct sum. We also consider ASM pattern avoidance from an algebro-geometric perspective.

  PublisherDOI

• Some algebraic properties of ASM varieties

  Glasgow Mathematical Journal · 2025-10-22

  articleOpen accessCorresponding

  Abstract Fulton’s matrix Schubert varieties are affine varieties that arise in the study of Schubert calculus in the complete flag variety. Weigandt showed that arbitrary intersections of matrix Schubert varieties, now called ASM varieties, are indexed by alternating sign matrices (ASMs), objects with a long history in enumerative combinatorics. It is very difficult to assess Cohen–Macaulayness of ASM varieties or to compute their codimension, though these properties are well understood for matrix Schubert varieties due to work of Fulton. In this paper, we study these properties of ASM varieties with a focus on the relationship between a pair of ASMs and their direct sum. We also consider ASM pattern avoidance from an algebro-geometric perspective.

  PublisherOA PDFDOI

• Polarization and Gorenstein liaison

  Journal of the London Mathematical Society · 2025-12-01

  articleOpen accessCorresponding

  Abstract A major open question in the theory of Gorenstein liaison is whether or not every arithmetically Cohen–Macaulay subscheme of can be G‐linked to a complete intersection. Migliore and Nagel showed that if such a scheme is generically Gorenstein (e.g., reduced), then, after re‐embedding so that it is viewed as a subscheme of , indeed it can be G‐linked to a complete intersection. Motivated by this result, we consider techniques for constructing G‐links on a scheme from G‐links on a closely related reduced scheme. Polarization is a tool for producing a squarefree monomial ideal from an arbitrary monomial ideal. Basic double G‐links on squarefree monomial ideals can be induced from vertex decompositions of their Stanley–Reisner complexes. Given a monomial ideal and a vertex decomposition of the Stanley–Reisner complex of its polarization , we give conditions that allow for the lifting of an associated basic double G‐link of to a basic double G‐link of itself. We use the relationship we develop in the process to show that the Stanley–Reisner complexes of polarizations of stable Cohen– Macaulay monomial ideals are vertex decomposable. We then introduce and study polarization of a Gröbner basis of an arbitrary homogeneous ideal and give a relationship between geometric vertex decomposition of a polarization and elementary G‐biliaison that is analogous to our result on vertex decomposition and basic double G‐linkage.

  PublisherDOI

• Algebra and geometry of ASM weak order

  ArXiv.org · 2025-02-26

  preprintOpen access

  Much of modern Schubert calculus is centered on Schubert varieties in the complete flag variety and on their classes in its integral cohomology ring. Under the Borel isomorphism, these classes are represented by distinguished polynomials called Schubert polynomials, introduced by Lascoux and Schützenberger. Knutson and Miller showed that Schubert polynomials are multidegrees of matrix Schubert varieties, affine varieties introduced by Fulton, which are closely related to Schubert varieties. Many roads to studying Schubert polynomials pass through unions and intersections of matrix Schubert varieties. The third author showed that the natural indexing objects of arbitrary intersections of matrix Schubert varieties are alternating sign matrices (ASMs). Every ASM variety is expressible as a union of matrix Schubert varieties. Many fundamental algebro-geometric invariants (e.g., codimension, degree, and Castelnuovo--Mumford regularity) are well understood combinatorially for matrix Schubert varieties, substantially via the combinatorics of strong Bruhat order on $S_n$. The extension of strong order to ASM(n), the set of $n \times n$ ASMs, has so far not borne as much algebro-geometric fruit for ASM varieties. Hamaker and Reiner proposed an extension of weak Bruhat order from $S_n$ to ASM(n), which they studied from a combinatorial perspective. In the present paper, we place this work on algebro-geometric footing. We use weak order on ASMs to give a characterization of codimension of ASM varieties. We also show that weak order operators commute with K-theoretic divided difference operators and that they satisfy the same derivative formula that facilitated the first general combinatorial computation of Castelnuovo--Mumford regularity of matrix Schubert varieties. Finally, we build from these results to generalizations that apply to arbitrary unions of matrix Schubert varieties.

  PublisherOA PDFDOI

• Multicomplex Configurations: a case study in Gorenstein Liaison

  ArXiv.org · 2025-07-14

  preprintOpen access1st authorCorresponding

  We introduce and investigate multicomplex configurations, a class of projective varieties constructed via specialization of the polarizations of Artinian monomial ideals. Building upon geometric polarization and geometric vertex decomposition, we establish conditions under which such configurations retain desirable algebraic properties. In particular, we show that, given suitable choices of linear forms for substitution, the resulting ideals admit Gröbner bases with prescribed initial ideals and are in the Gorenstein liaison class of a complete intersection.

  PublisherOA PDFDOI

• The MatrixSchubert package for Macaulay2

  Journal of Software for Algebra and Geometry · 2025-05-20

  articleOpen access

  We introduce the MatrixSchubert package for the computer algebra system Macaulay2.This package has tools to construct and study matrix Schubert varieties and alternating sign matrix (ASM) varieties.The package also introduces tools for quickly computing homological invariants of such varieties, finding the components of an ASM variety, and checking if a union of matrix Schubert varieties is an ASM variety.

  PublisherOA PDFDOI

• Polarization and Gorenstein liaison

  arXiv (Cornell University) · 2024-06-28

  preprintOpen access

  A major open question in the theory of Gorenstein liaison is whether or not every arithmetically Cohen--Macaulay subscheme of $\mathbb{P}^n$ can be G-linked to a complete intersection. Migliore and Nagel showed that, if such a scheme is generically Gorenstein (e.g., reduced), then, after re-embedding so that it is viewed as a subscheme of $\mathbb{P}^{n+1}$, indeed it can be G-linked to a complete intersection. Motivated by this result, we consider techniques for constructing G-links on a scheme from G-links on a closely related reduced scheme. Polarization is a tool for producing a squarefree monomial ideal from an arbitrary monomial ideal. Basic double G-links on squarefree monomial ideals can be induced from vertex decompositions of their Stanley--Reisner complexes. Given a monomial ideal $I$ and a vertex decomposition of the Stanley--Reisner complex of its polarization $P(I)$, we give conditions that allow for the lifting of an associated basic double G-link of $P(I)$ to a basic double G-link of $I$ itself. We use the relationship we develop in the process to show that the Stanley--Reisner complexes of polarizations of stable Cohen--Macaulay monomial ideals are vertex decomposable. We then introduce and study polarization of a Gröbner basis of an arbitrary homogeneous ideal and give a relationship between geometric vertex decomposition of a polarization and elementary G-biliaison that is analogous to our result on vertex decomposition and basic double G-linkage.

  PublisherOA PDFDOI

• Simplicial complexes and matroids with vanishing $T^2$

  arXiv (Cornell University) · 2024-06-04

  preprintOpen access

  We investigate quotients by radical monomial ideals for which $T^2$, the second cotangent cohomology module, vanishes. The dimension of the graded components of $T^2$, and thus their vanishing, depends only on the combinatorics of the corresponding simplicial complex. We give both a complete characterization and a full list of one dimensional complexes with $T^2=0$. We characterize the graded components of $T^2$ when the simplicial complex is a uniform matroid. Finally, we show that $T^2$ vanishes for all matroids of corank at most two and conjecture that all connected matroids with vanishing $T^2$ are of corank at most two.

  PublisherOA PDFDOI

• ON BASIC DOUBLE G-LINKS OF SQUAREFREE MONOMIAL IDEALS

  Journal of Commutative Algebra · 2024-05-16

  article1st authorCorresponding

  Nagel and Römer introduced the class of weakly vertex decomposable simplicial complexes, which include matroid, shifted, and Gorenstein complexes as well as vertex decomposable complexes. They proved that the Stanley–Reisner ideal of every weakly vertex decomposable simplicial complex is Gorenstein linked to an ideal of indeterminates via a sequence of basic double G-links. In this paper, we explore basic double G-links between squarefree monomial ideals beyond the weakly vertex decomposable setting. Our first contribution is a structural result about certain basic double G-links which involve an edge ideal. Specifically, suppose I(G) is the edge ideal of a graph G. When I(G) is a basic double G-link of a monomial ideal B on an arbitrary homogeneous ideal A, we give a generating set for B in terms of G and show that this basic double G-link must be of degree 1. Our second focus is on examples from the literature of simplicial complexes known to be Cohen–Macaulay but not weakly vertex decomposable. We show that these examples are not basic double G-links of any other squarefree monomial ideals.

  PublisherDOI

Frequent coauthors

• Christine Berkesch

  8 shared

• C-Y. Jean Chan

  6 shared

• Janet Page

  5 shared

• Janet Vassilev

  University of New Mexico

  5 shared

• Laura Felicia Matusevich

  Texas A&M University

  5 shared

• Jenna Rajchgot

  5 shared

• Alexander Blose

  University of Kentucky

  3 shared

• Anna Weigandt

  University of Minnesota

  3 shared