Research Interests

Schubert Calculus

Schur polynomials have a well known combinatorial interpretation as a sum over semistandard tableaux. Schubert polynomials lift the Schur polynomials, so it is natural to ask for a similar tableau like formula. With Alexander Yong, I introduced the prism tableau model for Schubert polynomials. Each prism tableau is a tuple of semistandard tableaux. We show how to obtain each Schubert polynomial as a weighted sum over prism tableaux.

Prism tableaux are closely related to poset of alternating sign matrices. For general prism shapes, the corresponding polynomial is a multiplicity free sum of Schubert polynomials. These terms have an explicit order theoretic description using the lattice of alternating sign matrices. As part of this work, I studied Rothe diagrams for ASMs and gave a graphical description of the essential set of an ASM.

I have also studied Rothe diagram of ASMs in connection with the bumpless pipe dreams of Lam-Lee-Shimozono. There is a natural bijection between (possibly non-reduced) square bumpless pipe dreams and ASMs. This gives an explicit connection between the LLS formula for double Schubert polynomials and Lascoux's formula for double Grothendieck polynomials. Lascoux's formula is stated in terms of the key of an ASM. I have shown that the pipe dream interpretation of ASMs produces a new method to obtain the key which is analogous to the theory of ordinary pipe dreams.

Combinatorial Algebraic Geometry

Matrix Schubert varieties are defined by imposing rank conditions on certain northwest submatrices of a generic matrix. Fulton showed that matrix Schubert varieties are irreducible if and only if their rank conditions are defined by a permutation matrix. Knutson and Miller gave a geometric interpretation for the pipe dream formula for Schubert polynomials via Gröbner degenerations of matrix Schubert varieties. They showed for any antidiagonal term order, pipe dreams naturally label the components of the limit variety. In work with Patricia Klein, I established a similar Gröbner geometric interpretation for bumpless pipe dreams. Namely, we show there exist diagonal term orders, for which bumpless pipe dreams identify the irreducible components of the limit scheme, with multiplicity. Initial progress towards this result was made in joint work with Zachary Hamaker and Oliver Pechenik.

Alternating sign matrix varieties generalize matrix Schubert varieties. Geometrically, they are unions of matrix Schubert varieties. The containment order on ASM varieties is equivalent to the lattice structure on ASMs. I used combinatorial commutative algebra to study the multidegrees of ASM varieties. It is well known that for matrix Schubert varieties these multidegrees are Schubert polynomials. This follows from work of Fulton, Knutson-Miller, and Fehér-Rimányi. The prism tableau formula for alternating sign matrices provides a combinatorial formula for the multidegree of a general ASM variety. Furthermore, prism tableaux naturally label the Stanley-Reisner complex of the initial scheme of an ASM variety under an antidiagonal Gröbner degeneration.

Castelnuovo-Mumford Regularity

The Castelnuovo-Mumford regularity of a graded module gives a measure of the complexity of its minimal free resolution. I have been studying Castelnuovo-Mumford regularity of generalized determinantal ideals, seeking explicit combinatorial interpretations of this statistic. For coordinate rings of matrix Schubert varieties, this problem was resolved completely in joint work with Oliver Pechenik and David Speyer. I also studied tableau based formulas for Grassmannian matrix Schubert varieties, as well as the vexillary and 1432 permutation pattern avoiding cases. The vexillary case allows for study of certain Kazhdan-Lusztig varieties.

Quiver Representations

Reineke studied a family of quantum dilogarithm identities, each defined by a Dynkin quiver. With Alexander Yong and Richárd Rimányi, I gave an elementary proof of Reineke's identities in type A. We established a related identity bijectively, using generating series for partitions. This new identity recovers Reineke's identity. The proof uses a generalization of Durfee squares. Future work in this direction includes proving the type D and E identities using generating series and studying related dilogarithm identities.