Publications and Preprints

Castelnuovo-Mumford regularity of ladder determinantal varieties and patches of Grassmannian Schubert varieties

With Jenna Rajchgot and Colleen Robichaux.
J. Algebra 617 (2023), 160-191.

We give diagrammatic degree formulas for Grothendieck polynomials indexed by vexillary and 2143-avoiding permutations. We apply the vexillary formula to compute Castelnuovo-Mumford regularity one-sided mixed ladder determinantal ideals and of classes of Kazhdan-Lusztig ideals.

A dual Littlewood-Richardson rule and extensions

With Oliver Pechenik.
Preprint, arXiv:2202.111851.

Wyser (2013) gave a rule in terms of (p,q)-clans for multiplying certain Schubert classes indexed by inverse Grassmannian permutations. We introduce backstable clans and extend Wyser's rule to arbitrary pairs of inverse Grassmannian permutations. Using differential operators, we prove linear relations among Schubert structure coefficients. We use these relations to derive new, multiplicity free rules for multiplying inverse Grassmannian classes by classes indexed by permutations that are left weak order covered by inverse Grassmannian permutations.

Castelnuovo-Mumford regularity of matrix Schubert varieties

With Oliver Pechenik and David E Speyer.
Selecta Mathematica, to appear, 2024, 37 pages, arXiv:2111.10681.

We give a formula for the regularity of matrix Schubert varieties in terms of a simple permutation statistic.

Bumpless pipe dreams encode Gröbner geometry of Schubert polynomials

With Patricia Klein.
Preprint, arXiv:2108.08370.

We find diagonal degenerations of matrix Schubert varieties whose irreducible components are counted by bumpless pipe dreams, with multiplicity.

Gröbner geometry of Schubert polynomials through ice

With Zachary Hamaker and Oliver Pechenik.
Adv. Math. 398 (2022), 108228.

We study the diagonal Gröbner geometry of matrix Schubert varieties and conjecture a combinatorial connection to bumpless pipe dreams.

Bumpless pipe dreams and alternating sign matrices

J. Combin. Theory Ser. A, 182 (2021), 105470.

I explain how to derive the formula of Lam, Lee, and Shimozono for double Schubert polynomials by reinterpreting an older formula of Lascoux.

Degrees of symmetric Grothendieck polynomials and Castelnuovo-Mumford regularity

With Jenna Rajchgot, Yi Ren, Colleen Robichaux, and Avery St. Dizier.
Proc. Amer. Math. Soc. 149 (2021), 1405-1416.

We give an explicit formula for the degrees of symmetric Grothendieck polynomials. This allows us to compute the regularity of Grassmannian matrix Schubert varieties.

Derivatives of Schubert polynomials and proof of a determinant conjecture of Stanley

With Zachary Hamaker, Oliver Pechenik, and David E Speyer.
Algebr. Comb. 3 (2020), no. 2, 301--307.

We prove a new derivative identity for Schubert polynomials. Using this, we give a simple proof of Macdonald's identity and solve a determinant conjecture of Stanley.

Prism tableaux for alternating sign matrix varieties

Preprint, arXiv:1708.07236.

We characterize the subvarieties of a matrix space which are defined by northwest rank conditions in terms of alternating sign matrices. We use prism tableaux to give a formula for their multidegrees.

Schubert polynomials, 132-patterns, and Stanley's conjecture

Algebr. Comb. 1 (2018), no. 4, 415-423.

We prove a conjecture of Stanley, characterizing two term Schubert polynomials.

Partition identities and quiver representations

With Richard Rimanyi and Alexander Yong.
J. Algebraic Combin. 47 (2018), no. 1, 129-169.

We give new proofs of Reineke's quantum dilogarithm identities using a generalized Durfee square identity.

The Prism tableau model for Schubert polynomials

With Alexander Yong.
J. Combin. Theory Ser. A. 154 (2018), 551-582.

We present a new model for Schubert polynomials which generalizes the tableau model for Schur polynomials.

Constraint consensus methods for finding interior feasible points in second-order cones.

With Kaitlyn Tuthill and Shafiu Jibrin.
J. Appl. Math. (2010), Art. ID 307209, 19 pp.

We give an back-tracking line search algorithm for finding the feasible region of systems of inequalities defined by second-order cones.

Other Writing

Bumpless pipe dreams encode Gröbner geometry of Schubert polynomials

With Patricia Klein
To appear in the Proceedings of the 34th International Conference on Formal Power Series and Algebraic Combinatorics (2022).

FPSAC extended abstract. Accepted as a talk.

Regularity of Matrix Schubert Varieties

With Oliver Pechenik and David E Speyer
To appear in the Proceedings of the 34th International Conference on Formal Power Series and Algebraic Combinatorics (2022).

FPSAC extended abstract. Accepted as a poster.

Gröbner Geometry of Schubert Polynomials Through Ice

With Zachary Hamaker and Oliver Pechenik
Proceedings of the 33rd International Conference on "Formal Power Series and Algebraic Combinatorics", January 10 - 13, 2022, Bar Ilan University, Ramat Gan, Israel (2021).

FPSAC extended abstract. Accepted as a talk.

Prism tableaux for alternating sign matrix varieties

Proceedings of the 30th International Conference on "Formal Power Series and Algebraic Combinatorics", July 16 - 20, 2018, Dartmouth College, Hanover, USA. Sém. Lothar. Combin. 80B, Art. 52, 12 pp. (2018).

FPSAC extended abstract. Accepted as a poster.

The Prism tableau model for Schubert polynomials

With Alexander Yong
28th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2016), 1203–1214, Discrete Math. Theor. Comput. Sci. Proc., BC (2016).

FPSAC extended abstract. Accepted as a poster.