The main part of my research is in the intersection of combinatorics and representation theory. The motivation for the problems I am inerested in usually come from mathematical physics, more specifically with integrable systems and (Drinfel'd–Jimbo) quantum groups. My research also has motivations and applications in algebraic geometry.
My focus is in crystals, which is a combinatorial realization of a special basis for a quantum group representation. Crystals encode the representation through certain edge-colored weighted directed graphs known as crystal graphs. They also translate many of the algebraic properties of the representations into combinatorial rules such as tensor products, dualities, and restrictions. This can also be used to give representation theoretic interpretations of more classical combinatorial maps. For instance, in type \(A_n\), crystals are given by semistandard tableaux and maps such as evacuation, promotion, (co)plactic operations have natural interpretations using crystals.
For more information on crystals (in particular, how you can use them in SageMath), see the SageMath thematic tutorial on Lie methods and related combinatorics. A good book for crystals is Crystal bases: Representations and Combinatorics by Danial Bump and Anne Schilling.
One important class of crystals are certain finite crystals for affine type called Kirillov–Reshetikhin (KR) crystals, which are the crystal bases of Kirillov–Reshetikhin modules. These are no longer highest weight crystals, and so present unique challenges outside of the general theory. Despite this, they are known to be connected through their character theory with (nonsymmetric) Macdonald polynomials specialized at \(t = 0\), Q-systems, integrable systems, and soliton cellular automata. They are also related to Demazure subcrystals of affine highest weight crystals. Open problems include showing the existence of KR crystals for the exceptional affine types, constructing combinatorial model for the KR crystals, and describing a uniform model for all KR crystals.
One combinatorial model that I have been working on is given by rigged configurations. Rigged configurations come from mathematical physics, where they were originally used to index solutions to the Bethe ansatz to Heisenberg spin chains. The \(X=M\) conjecture of Hatayama et al. says there is an equality of the weighted generating functions of rigged configurations and classically highest weight elements in a tensor product of KR crystals. I have been working on the program to prove the \(X=M\) conjecture by constructing a particular recursive bijection \(\Phi\). Masato Okado, Anne Schilling and I have shown this in all nonexceptional affine types, and I am currently working on proving such a bijection in the exceptional affine types. The bijection \(\Phi\) can also be considered as a linearization of the dynamics of solution cellular automata, a generalization of the Takahashi–Satsuma box-ball system, which have natural interpretations using rigged configurations.
From \(\Phi\), we can translate the natural (classical) crystal structure from the Kirillov–Reshetikhin crystals to rigged configurations. This was first done for highest weight crystals \(B(\lambda)\) in simply-laced types by Anne Schilling in 2006. Ben Salisbury and I then extended this to all (symmetrizable) types, Borcherds (or generalized Kac–Moody Lie) algebras, and to the crystal \(B(\infty)\). It appears that rigged configurations are a natural crystal model as the \(*\)-involution corresponds to the natural involution of replacing riggings with coriggings.
One of my major research projects is constructing an analog of this theory for the affine Lie superalgebras.
Another major research project has been developing a K-theory extension of type \(A_n\) crystals. Schur functions are polynomial representatives of Schubert classes in the cohomology ring of the Grassmannian, the set of \(k\)-dimensional planes in \(\mathbb{C}^n\). If we instead consider the K-theory of the Grassmannian, polynomial representatives are the symmetric Grothendieck polynomials. Symmetric Grothendieck polynomials are the generating functions of set-valued tableaux, a semistandard tableau where entries can be nonempty sets of positive integers. Cara Monical, Oliver Pechenik, and I constructed a type \(A_n\) crystal structure on set-valued tableaux, and using the divided difference operator definition, proposed a new structure that will categorify symmetric Grothendieck polynomials that we call a K-crystal. We have currently shown this structure exists for all rectangular shapes. Recently, I have been using colored integrable lattice models to prove various combinatorial properties of (symmetric) Grothendieck polynomials.
Other things I am working on or interested in:
- Coxeter groups: These include the symmetric group and Weyl groups.
- Artin groups: These include the braid group.
- Complex reflection groups: Coxeter groups correspond to the case of real reflections.
- Symmetric functions (see also the ring of symmetric functions): the universal character ring of type \(A_n\) crystals. See also the SageMath tutorial.