The main part of my research is in the intersection of combinatorics and representation theory. The motivations for the problems I am inerested in usually come from mathematical physics, more specifically with integrable systems and (Drinfel'd–Jimbo) quantum groups.

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 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,
and T-systems. They are also related to Demazure subcrystals of affine highest
weight 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 exists a bijection between 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 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. We are also currently extending this crystal structure to Borcherds (or generalized Kac–Moody Lie) algebras.

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.