RESEARCH OUTLINE
The theory of Kashiwara’s crystal bases gives combinatorial models for representations of symmetrizable Kac–Moody
Lie algebras, which has led to deep connections to other areas of mathematics and physics. One important connection is tostatistical mechanics, where crystals arise in the study of integrable systems, such as solvable lattice models and quantum
spin chains, with a particular emphasis the crystal bases associated to Kirillov–Reshetikhin (KR) modules. Recently, I
have been generalizing of crystals to areas such as K-theoretic Schubert calculus.
Circa 2000, G. Hatayama et al. [4,5] conjectured a generalization of the work of Kerov, Kirillov, and Reshetikhin abouta generating series identity relating the representations of affine quantum groups with the XXZ Heisenburg spin chain,
which is now known as the X = M conjecture. To describe this conjecture, we first note that KR modules are a certainclass of nice finite dimensional representations of affine quantum groups that have been well-studied by numerous authors.
One outstanding conjecture regarding KR modules has been they admit crystal bases, which has been solved except fora few nodes in exceptional types by myself and K. Naoi [11]. The X = M conjecture can be proven by constructing abijection between a combinatorial object called rigged configurations and the elements of a tensor product of KR crystals.
I have conjectured a bijection in all affine types, which I have been proven in all nonexceptional types with M. Okado andA. Schilling [14] and numerous special cases in the exceptional types [16, 17]. I have applied this bijection to the study ofsoliton cellular automata [8] (Xuan Liu was then an undergraduate), a generalization of the Takahashi–Satsuma box-ballsystem (an ultradiscrete version of the Korteweg–de Vries (KdV) equation). In a series of papers with B. Salisbury, we
have been studying the crystal structure of rigged configurations and generalizing it to all Kac–Moody Lie algebras andBorcherds algebras [15], where we show they are a natural model for crystals. E, Aas, D. Grinberg, and I also used a KR
crystal interpretation of the probabilistic model TASEP on a ring in [1] to prove some conjectures.
Beyond the X = M conjecture, KR modules generate an interesting class of affine quantum group representations,which D. Hernandez and B. Leclerc conjecture categorifies a cluster algebra [7]. One approach to proving this conjecture
involves studying the R-matrix, the morphism that interchanges the tensor product of two KR modules. S.-j. Oh and I
have studied [12,13] when the R-matrix is not an isomorphism, which gives a relation between KR modules called Dorey’srule, and can be applied to prove the categorification conjecture.
Schubert calculus can be described as the study of the Grassmannian, the set of k-dimensional planes in n-dimensional
space. The cohomology ring and those for the full flag variety, which is the quotient of invertible n×n matrices by invertible
upper triangular matrices, has been well-studied by relating them to symmetric functions and basis for all polynomials,
respectively by using certain subvarities called Schubert varieties. We enrich this theory by instead looking at the K-theory
ring, which yields Grothendieck polynomials. For the Grassmannian case, these become symmetric functions and havenatural generalizations, all of which can be described combinatorially. I and coauthors show the associated combinatorics
admit crystal structures [6,9] and we conjecture a generalization of crystals in an effort to categorify the ring structure [9].
One natural generalization was recently shown by me and K. Motegi to be related with a random matrix process [10].
The R-matrix and KR modules can also be used as a simple model for ice called the six-vertex model, a prototypicalexample of a solvable lattice model. We can draw this diagrammatically as a square grid with labelings on edges thatsatisfy local conditions around each vertex and obtain generating functions describing the system using the Yang–Baxterequations. I and coauthors utilize this approach to understand Grothendieck polynomials and their “partial” versions
called Lascoux polynomials [2, 3], giving the first proven combinatorial rule for Lascoux polynomials.
References[1] E. Aas, D. Grinberg, and T. Scrimshaw. Multiline queues with spectral parameters. Comm. Math. Phys., 374(3):1743–1786, 2019.
[2] V. Buciumas and T. Scrimshaw. Double Grothendieck polynomials and colored lattice models. Int. Math. Res. Not. IMRN, 2020. Art. ID
rnaa327.
[3] V. Buciumas, T. Scrimshaw, and K. Weber. Colored five-vertex models and Lascoux polynomials and atoms. J. Lond. Math. Soc.,
102(3):1047–1066, 2020.
[4] G. Hatayama, A. Kuniba, M. Okado, T. Takagi, and Y. Yamada. Remarks on fermionic formula. In Recent developments in quantum affine
algebras and related topics (Raleigh, NC, 1998), volume 248 of Contemp. Math., pages 243–291. Amer. Math. Soc., Providence, RI, 1999.
[5] G. Hatayama, A. Kuniba, M. Okado, T. Takagi, and Z. Tsuboi. Paths, crystals and fermionic formulae. In MathPhys odyssey, 2001,
volume 23 of Prog. Math. Phys., pages 205–272. Birkhäuser Boston, Boston, MA, 2002.
[6] G. Hawkes and T. Scrimshaw. Crystal structures for canonical Grothendieck functions. Algebraic Combin., 3(3):727–755, 2020.
[7] D. Hernandez and B. Leclerc. A cluster algebra approach to q-characters of Kirillov–Reshetikhin modules. J. Eur. Math. Soc. (JEMS),
18(5):1113–1159, 2016.
[8] X. Liu and T. Scrimshaw. A uniform approach to soliton cellular automata using rigged configurations. Ann. Henri Poincaré, 20(4):1175–1215, 2019.
[9] C. Monical, O. Pechenik, and T. Scrimshaw. Crystal structures for symmetric Grothendieck polynomials. Transform. Groups, 2020. Toappear, doi:10.1007/s00031-020-09623-y.
[10] K. Motegi and T. Scrimshaw. Refined dual Grothendieck polynomials, integrability, and the Schur measure. Preprint, arXiv:2012.15011,2020.
[11] K. Naoi and T. Scrimshaw. Existence of Kirillov–Reshetikhin crystals for near adjoint nodes in exceptional types. J. Pure Appl. Algebra,
225(5):106593, 2021.
[12] Se-jin Oh and T. Scrimshaw. Categorical relations between Langlands dual quantum affine algebras: Exceptional cases. Comm. Math.
Phys., 368(1):295–367, 2019.
[13] S.-j. Oh and T. Scrimshaw. Simplicity of tensor products of Kirillov–Reshetikhin modules: nonexceptional affine and G types. Preprint,
arXiv:1910.10347, 2019.
[14] M. Okado, A. Schilling, and T. Scrimshaw. Rigged configuration bijection and proof of the X = M conjecture for nonexceptional affinetypes. J. Algebra, 516:1–37, 2018.
[15] B. Salisbury and T. Scrimshaw. Rigged configurations for generalized Kac–Moody algebras. Preprint, arXiv:1812.07746, 2018.
(3)[16] T. Scrimshaw. A crystal to rigged configuration bijection and the filling map for type D4 . J. Algebra, 448C:294–349, 2016.
[17] T. Scrimshaw. Uniform description of the rigged configuration bijection. Selecta Math. (N.S.), 26(3):article 42, 2020.