Peer reviewed research articles:

Geometric vertex decomposition and liaison for toric ideals of graphs (with Mike Cummings, Sergio Da Silva, and Adam Van Tuyl)

Algebraic Combinatorics, Volume 6 (2023) p. 965997

arXiv

Summary
The geometric vertex decomposability property for polynomial ideals is an idealtheoretic generalization of the vertex decomposability property for simplicial complexes. Indeed, a homogeneous geometrically vertex decomposable ideal is radical and CohenMacaulay, 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 squarefree degeneration with respect to a lexicographic order are geometrically vertex decomposable. As evidence, we prove the conjecture in the case that the universal Gröbner 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.
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CastelnuovoMumford regularity of ladder determinantal varieties and patches of Grassmannian Schubert varieties (with Colleen Robichaux and Anna Weigandt)
 Journal of Algebra
Volume 617, 1 March 2023, 160191

arXiv

Summary
We give degree formulas for Grothendieck polynomials indexed by vexillary permutations and 1432avoiding permutations via tableau combinatorics. These formulas generalize a formula for degrees of symmetric Grothendieck polynomials which appeared in previous joint work of the authors with Y. Ren and A. St. Dizier. We apply our formulas to compute CastelnuovoMumford regularity of classes of generalized determinantal ideals. In particular, we give combinatorial formulas for the regularities of all onesided mixed ladder determinantal ideals. We also derive formulas for the regularities of certain KazhdanLusztig ideals, including those coming from open patches of Schubert varieties in Grassmannians. This provides a correction to a conjecture of KumminiLakshmibaiSastrySeshadri (2015).
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Geometric vertex decomposition and liaison (with Patricia Klein)
 Forum of Math, Sigma, Volume 9, 2021, e70.
 arXiv
Summary
Geometric vertex decomposition and liaison are two frameworks that have
been used to produce similar results about similar families of algebraic varieties. In this paper, we establish an explicit connection between these approaches. In particular, we show that each geometrically vertex decomposable ideal is linked by a sequence of elementary Gbiliaisons of height 1 to an ideal of indeterminates and, conversely, that every Gbiliaison of a certain type gives rise to a geometric vertex decomposition. As a consequence, we can immediately conclude that several wellknown families of ideals are glicci.
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 An extended abstract of this paper was accepted to FPSAC 2021.

Type D quiver representation varieties, double Grassmannians, and symmetric varieties (with Ryan Kinser)
 Advances in Math. 376 (2021), 107454, 44 pp.
 arXiv
Summary
We consider three families of varieties: type D quiver representation varieties, double Grassmannians, and certain symmetric varieties. By constructing nice embeddings, we complete a circle of links which allows for translation of results about singularities of orbit closures, combinatorics of orbit closure containment, and torus equivariant Ktheory between these three families of varieties.
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Degrees of symmetric Grothendieck polynomials and CastelnuovoMumford regularity (with Yi Ren, Colleen Robichaux, Avery St. Dizier, Anna Weigandt)
 Proceedings of the AMS. 149 (2021), no. 4, 1405–1416.
 arXiv
Summary
We give an explicit formula for the degree of the Grothendieck polynomial of a Grassmannian permutation and a closely related formula for the CastelnuovoMumford regularity of the Schubert determinantal ideal of a Grassmannian permutation. We then provide a counterexample to a conjecture of KumminiLakshmibaiSastrySeshadri on a formula for regularities of standard open patches of particular Grassmannian Schubert varieties and show that our work gives rise to an alternate explicit formula in these cases. We end with a new conjecture on the regularities of standard open patches of arbitrary Grassmannian
Schubert varieties.
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Some combinatorial cases of the three matrix analog of Gerstenhaber's theorem (with Matthew Satriano and Wanchun Shen)

Three combinatorial formulas for type A quiver polynomials and Kpolynomials (with Ryan Kinser and Allen Knutson)

Duke Math. J.
Volume 168, Number 4 (2019), 505551.
 arXiv

Summary
We provide combinatorial formulas for the multidegree and Kpolynomial of an (arbitrarily oriented) type A quiver locus embedded inside of its representation space. These formulas are generalizations of three of KnutsonMillerShimozono's formulas from the equioriented setting:
 The ratio formulas express each Kpolynomial as a ratio of specialized double Grothendieck polynomials, and each multidegree as a ratio of specialized double Schubert polynomials.
 The pipe formulas express each Kpolynomial as an alternating sum over pipe dreams that fit inside of a particular shape, and each multidegree as a positive sum over reduced pipe dreams that fit inside of that same shape.
 The component formulas express each Kpolynomial as an alternating sum of products of Grothendieck polynomials, and each multidegree as a positive sum of products of Schubert polynomials. The summands are indexed by lacing diagrams associated to the type A quiver locus. The Kpolynomial component formula was first conjectured by BuchRimànyi, and the multidegree component formula was first proved by BuchRimànyi.
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New classes of examples satisfying the three matrix analog of Gerstenhaber's theorem (with Matthew Satriano)

Journal of Algebra, Volume 516, 15 December 2018, Pages 245270.
 arXiv
Summary
In 1961, Gerstenhaber proved the following theorem: if k is a field and X and Y are commuting dxd matrices with entries in k, then the unital kalgebra generated by these matrices has dimension at most d. The analog of this statement for four or more commuting matrices is false. The three matrix version remains open. We use commutativealgebraic techniques to prove that the three matrix analog of Gerstenhaber's theorem is true for some new classes of examples.
In particular, we translate this three commuting matrix statement into an equivalent statement about certain maps between modules, and prove that this commutativealgebraic reformulation is true in special cases. We end with ideas for an inductive approach intended to handle the three matrix analog of Gerstenhaber's theorem more generally.
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Lower bound cluster algebras: presentations, CohenMacaulayness, and normality (with Greg Muller and Bradley Zykoski)

Algebraic Combinatorics, Volume 1 (2018) no. 1, p. 95114.
 arXiv

Summary
Lower bound cluster algebras are approximations to cluster algebras which consider only those cluster variables one mutation away from an initial seed. In this paper, we give an explicit presentation for each lower bound cluster algebra. Using these presentations, we show that each lower bound algebra Gröbner degenerates to the StanleyReisner scheme of a vertexdecomposable ball or sphere, and is thus CohenMacaulay. Finally, we use StanleyReisner combinatorics and a result of KnutsonLamSpeyer to show that all lower bound algebras are normal.
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Matrix Schubert varieties and Gaussian conditional independence models (with Alex Fink and Seth Sullivant)

Journal of Algebraic Combinatorics. December 2016, Volume 44, Issue 4, pp 10091046.
 arXiv
Summary
Here we study matrix Schubert varieties, and their analogs for symmetric and upper triangular matrices, in view of two applications to algebraic statistics: we observe that special conditional independence models for Gaussian random variables are intersections of symmetric matrix Schubert varieties, and consequently obtain a combinatorial primary decomposition algorithm for some conditional independence ideals. We also characterize the vanishing ideals of Gaussian graphical models for generalized Markov chains.
In the course of this investigation, we are led to consider three related stratifications, which come from the Schubert stratification of a flag variety. We provide some combinatorial results, including describing the stratifications using the language of rank arrays and enumerating the strata in each case.
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Singularities of locally acyclic cluster algebras (with Angélica Benito, Greg Muller, and Karen E. Smith)

Algebra & Number Theory 9 (2015), no. 4, 913  936.

arXiv

Summary
We investigate the singularities of cluster algebras using characteristic p>0 techniques. We prove that locally acyclic cluster algebras are strongly Fregular, and that every upper cluster algebra always has a natural Frobenius splitting. We also provide examples to show that not all upper cluster algebras are strongly Fregular if the local acyclicity is dropped.
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Type A quiver loci and Schubert varieties (with Ryan Kinser)

Journal of Commutative Algebra 7 (2015), no. 2, 265301.
 arXiv

Summary
We show that each quiver locus of an arbitrarily oriented type A quiver is isomorphic, up to smooth factor, to an open subvariety of a Schubert variety.
We consequently recover results of Bobinski and Zwara that quiver loci of type A quivers are normal, CohenMacaulay, and have rational singularities. We also see that quiver locus containment is determined by Bruhat order on the symmetric group, and that each representation space of a type A quiver admits a Frobenius splitting for which all of its quiver loci are compatibly Frobenius split.
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Preprints:

Gröbner geometry for regular nilpotent Hessenberg Schubert cells (with Mike Cummings, Sergio Da Silva, and Megumi Harada)

Hadamard Products and Binomial Ideals (by the 2021/2022 Combinatorial Algebraic Geometry research group at McMaster University: Busra Atar, Kieran Bhaskara, Adrian Cook, Sergio Da Silva, Megumi Harada, Jenna Rajchgot, Adam Van Tuyl, Runyue Wang, Jay Yang)

On basic double links of squarefree monomial ideals (with Patricia Klein and Matthew Koban)

arXiv

Summary
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 StanleyReisner ideal of every weakly vertex decomposable simplicial complex is Gorenstein linked to an ideal of indeterminates via a sequence of basic double links. In this paper, we explore basic double links between squarefree monomial ideals beyond the weakly vertex decomposable setting.
Our first contribution is a structural result about certain basic double 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 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 link must be of degree 1. Our second focus is on examples from the literature of simplicial complexes known to be CohenMacaulay but not weakly vertex decomposable. We show that these examples are not basic double links of any other squarefree monomial ideals.
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Toric and tropical Bertini theorems in positive characteristic (with Francesca Gandini, Milena Hering, Diane Maclagan, Fatemeh Mohammadi, Ashley K. Wheeler, and Josephine Yu)

arXiv

Summary
We generalize the toric Bertini theorem of Fuchs, Mantova, and Zannier to positive characteristic. A key part of the proof is a new algebraically closed field containing the field \kk(t_1,\dots,t_d) of rational functions over an algebraically closed field \kk of prime characteristic. As a corollary, we extend the tropical Bertini theorem of Maclagan and Yu to arbitrary characteristic, which removes the characteristic dependence from the dconnectivity result for tropical varieties from that paper.
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Gröbner bases, symmetric matrices, and type C KazhdanLusztig varieties (with Laura Escobar, Alex Fink , and Alexander Woo)

arXiv

Summary
We study a class of combinatoriallydefined polynomial ideals which are generated by minors of a generic symmetric matrix. Included within this class are the symmetric determinantal ideals, the symmetric ladder determinantal ideals, and the symmetric Schubert determinantal ideals of A. Fink, J. Rajchgot, and S. Sullivant. Each ideal in our class is a type C analog of a KazhdanLusztig ideal of A. Woo and A. Yong; that is, it is the schemetheoretic defining ideal of the intersection of a type C Schubert variety with a type C opposite Schubert cell, appropriately coordinatized. The KazhdanLusztig ideals that arise are exactly those where the opposite cell is 123avoiding. Our main results include Gröbner bases for these ideals, prime decompositions of their initial ideals (which are StanleyReisner ideals of subword complexes) and combinatorial formulas for their multigraded Hilbert series in terms of pipe dreams.
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Ph.D. Thesis:

Compatibly split subvarieties of the Hilbert scheme of points in the plane
 Ph.D. Thesis, Cornell University. Advisor: Allen Knutson.
 arXiv

Summary
We explicitly describe all compatibly split subvarieties of the Hilbert scheme of n points in the plane for small values of n, and we provide some partial results for arbitrary n. We then restrict to a specific affine open patch (for arbitrary n) and find all compatibly split subvarieties, their defining ideals, and Gröbner degenerations to StanleyReisner schemes. The associated simplicial complexes are used to prove that certain compatibly split subvarieties of this open patch are CohenMacaulay.
Also included in this document is some background material on the Hilbert scheme of points in the plane, Frobenius splitting, moment polyhedra (in the algebraic setting), and Gröbner bases/degenerations.
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Other:

Types A and D quiver representation varieties

Extended abstract for the Oberwolfach Reports (MiniWorkshop: Degeneration Techniques in Representation Theory, October 2019)
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