Research activity information

• A strange five vertex model and multispecies ASEP on a ring
  Atsuo Kuniba; Masato Okado; Travis Scrimshaw
  Algebraic Combinatorics, 06 Jan. 2026
  English, Scientific journal, We revisit the problem of constructing the stationary states of the multispecies asymmetric simple exclusion process on a one-dimensional periodic lattice. Central to our approach is a quantum oscillator weighted five vertex model which features a strange weight conservation distinct from the conventional one. Our results clarify the interrelations among several known results and refine their derivations. For instance, the stationary probability derived from the multiline queue construction by Martin (2020) and Corteel–Mandelshtam–Williams (2022) is identified with the partition function of a three-dimensional system. The matrix product operators by Prolhac–Evans–Mallick (2009) acquire a natural diagrammatic interpretation as corner transfer matrices (CTM). The origin of their recursive tensor structure, as questioned by Aggarwal–Nicoletti–Petrov (2023), is revealed through the CTM diagrams. Finally, the derivation of the Zamolodchikov–Faddeev algebra by Cantini–de Gier–Wheeler (2015) is made intrinsic by elucidating its precise connection to a solution to the Yang–Baxter equation originating from quantum group representations.
• Crystals for shifted key polynomials
  Eric Marberg; Travis Scrimshaw
  Algebras and Representation Theory, 28 Jul. 2025
  English, Scientific journal, Abstract
  This article continues our study of P- and Q-key polynomials, which are (non-symmetric) “partial” Schur P- and Q-functions as well as “shifted” versions of key polynomials. Our main results provide a crystal interpretation of P- and Q-key polynomials, namely, as the characters of certain connected subcrystals of normal crystals associated to the queer Lie superalgebra
  
  $$\mathfrak {q}_n$$
  
  
  q
  n
  
  
  
  . In the P-key case, the ambient normal crystals are the
  
  $$\mathfrak {q}_n$$
  
  
  q
  n
  
  
  
  -crystals studied by Grantcharov et al., while in the Q-key case, these are replaced by the extended
  
  $$\mathfrak {q}_n$$
  
  
  q
  n
  
  
  
  -crystals recently introduced by the first author and Tong. Using these constructions, we propose a crystal-theoretic lift of several conjectures about the decomposition of involution Schubert polynomials into P- and Q-key polynomials. We verify these generalized conjectures in a few special cases. Along the way, we establish some miscellaneous results about normal
  
  $$\mathfrak {q}_n$$
  
  
  q
  n
  
  
  
  -crystals and Demazure
  
  $$\mathfrak {gl}_n$$
  
  
  gl
  n
  
  
  
  -crystals.
• Limit Shapes and Fluctuations for $(GL_{n},GL_{k})$ Skew Howe Duality
  Dan Betea; Anton Nazarov; Pavel Nikitin; Travis Scrimshaw
  Annales Henri Poincaré, 21 Jul. 2025
  English, Scientific journal
• Refined dual Grothendieck polynomials, integrability, and the Schur measure
  Kohei Motegi; Travis Scrimshaw
  Selecta Mathematica, Jul. 2025
  English, Scientific journal
• Crystal invariant theory I: geometric RSK
  Benjamin Brubaker; Gabriel Frieden; Pavlo Pylyavskyy; Travis Scrimshaw
  Mathematische Zeitschrift, May 2025
  English, Scientific journal
• Smooth locus of twisted affine Schubert varieties and twisted affine Demazure modules
  Marc Besson; Jiuzu Hong
  Forum of Mathematics, Sigma, 2025
  English, Scientific journal, Abstract
  Let
  
  
  
  ${\mathscr {G } } $
  
  
  be a special parahoric group scheme of twisted type over the ring of formal power series over
  
  
  
  $\mathbb {C}$
  
  
  , excluding the absolutely special case of
  
  
  
  $A^{(2)}_{2\ell }$
  
  
  . Using the methods and results of Zhu, we prove a duality theorem for general
  
  
  
  ${\mathscr {G } } $
  
  
  : there is a duality between the level one twisted affine Demazure modules and the function rings of certain torus fixed point subschemes in affine Schubert varieties for
  
  
  
  ${\mathscr {G } } $
  
  
  . Along the way, we also establish the duality theorem for
  
  
  
  $E_6$
  
  
  . As a consequence, we determine the smooth locus of any affine Schubert variety in the affine Grassmannian of
  
  
  
  ${\mathscr {G } } $
  
  
  . In particular, this confirms a conjecture of Haines and Richarz.
• Free fermionic probability theory and K-theoretic Schubert calculus
  Shinsuke Iwao; Kohei Motegi; Travis Scrimshaw
  Forum of Mathematics, Sigma, 2025
  English, Scientific journal, Abstract
  
  For each of the four particle processes given by Dieker and Warren, we show the
  n
  -step transition kernels are given by the (dual) (weak) refined symmetric Grothendieck functions up to a simple overall factor. We do so by encoding the particle dynamics as the basis of free fermions first introduced by the first author, which we translate into deformed Schur operators acting on partitions. We provide a direct combinatorial proof of this relationship in each case, where the defining tableaux naturally describe the particle motions.
  
• Twisted multiline queues for the steady states of TASEP and TAZRP
  Olya Mandelshtam; Travis Scrimshaw
  Electronic Journal of Probability, 01 Jan. 2025
  Scientific journal
• Integrable systems and crystals for edge labeled tableaux
  Ajeeth Gunna; Travis Scrimshaw
  Journal of Algebra, Apr. 2024
  English, Scientific journal
• Free fermions and canonical Grothendieck polynomials
  Shinsuke Iwao; Kohei Motegi; Travis Scrimshaw
  Algebraic Combinatorics, 22 Feb. 2024
  English, Scientific journal
• Skew Howe duality and limit shapes of Young diagrams
  Anton Nazarov; Olga Postnova; Travis Scrimshaw
  Journal of the London Mathematical Society, Jan. 2024
  English, Scientific journal, AbstractWe consider the skew Howe duality for the action of certain dual pairs of Lie groups on the exterior algebra as a probability measure on Young diagrams by the decomposition into the sum of irreducible representations. We prove a combinatorial version of this skew Howe duality for the pairs , , , and using crystal bases, which allows us to interpret the skew Howe duality as a natural consequence of lattice paths on lozenge tilings of certain partial hexagonal domains. The ‐representation multiplicity is given as a determinant formula using the Lindström–Gessel–Viennot lemma and as a product formula. These admit natural ‐analogs that we show equals the ‐dimension of a ‐representation (up to an overall factor of ), giving a refined version of the combinatorial skew Howe duality. Using these product formulas (at ), we take the infinite rank limit and prove that the diagrams converge uniformly to the limit shape.
• On the structure and representation theory of q-deformed Clifford algebras
  Willie Aboumrad; Travis Scrimshaw
  Mathematische Zeitschrift, Jan. 2024
  English, Scientific journal
• Cellular subalgebras of the partition algebra
  Travis Scrimshaw
  Journal of Combinatorial Algebra, 21 Dec. 2023
  Scientific journal, We describe various diagram algebras and their representation theory using cellular algebras of Graham and Lehrer and the decomposition into half diagrams. In particular, we show the diagram algebras surveyed here are all cellular algebras and parameterize their cell modules. We give a new construction to build new cellular algebras from a general cellular algebra and subalgebras of the rook Brauer algebra that we call the cellular wreath product.
• K-theoretic crystals for set-valued tableaux of rectangular shapes
  Oliver Pechenik; Travis Scrimshaw
  Algebraic Combinatorics, 5, 3, 515, 536, Cellule {MathDoc}/{CEDRAM}, 07 Jul. 2022
  English, Scientific journal
• Candidate for the crystal B(−∞) for the queer Lie superalgebra
  Ben Salisbury; Travis Scrimshaw
  Kyoto Journal of Mathematics, 62, 2, Duke University Press, 01 Jun. 2022
  Scientific journal
• Kirillov–Reshetikhin crystals $B^{7,s}$ for type $E_7^{(1)}$
  Rekha Biswal; Travis Scrimshaw
  Communications in Algebra, 1, 16, Informa {UK} Limited, 08 Feb. 2022
  English, Scientific journal
• Quasi-solvable lattice models for and Demazure atoms and characters
  Valentin Buciumas; Travis Scrimshaw
  Forum of Mathematics, Sigma, 10, Cambridge University Press ({CUP}), 2022
  English, Scientific journal, Abstract
  We construct coloured lattice models whose partition functions represent symplectic and odd orthogonal Demazure characters and atoms. We show that our lattice models are not solvable, but we are able to show the existence of sufficiently many solutions of the Yang–Baxter equation that allow us to compute functional equations for the corresponding partition functions. From these functional equations, we determine that the partition function of our models are the Demazure atoms and characters for the symplectic and odd orthogonal Lie groups. We coin our lattice models as quasi-solvable. We use the natural bijection of admissible states in our models with Proctor patterns to give a right key algorithm for reverse King tableaux and Sundaram tableaux.
• Crystal structures for symmetric Grothendieck polynomials
  Cara Monical; Oliver Pechenik; Travis Scrimshaw
  Transformation Groups, 26, 3, 1025, 1075, Springer Science and Business Media {LLC}, Sep. 2021
  English, Scientific journal
• Rigged configurations and the ⁎-involution for generalized Kac–Moody algebras
  B. Salisbury; T. Scrimshaw
  Journal of Algebra, 573, 148, 168, Elsevier {BV}, May 2021
  English, Scientific journal
• Existence of Kirillov–Reshetikhin crystals for near adjoint nodes in exceptional types
  Katsuyuki Naoi; Travis Scrimshaw
  Journal of Pure and Applied Algebra, 225, 5, 106593, 106593, Elsevier {BV}, May 2021
  English, Scientific journal
• The regularity of almost-commuting partial Grothendieck--Springer resolutions and parabolic analogs of Calogero--Moser varieties
  Mee Seong Im; Travis Scrimshaw
  Journal of Lie Theory, 2021
  Scientific journal, Consider the moment map $\mu \colon T^*(\mathfrak{p} \times \mathbb{C}^n) \to \mathfrak{p}^*$ for a parabolic subalgebra $\mathfrak{p}$ of $\mathfrak{gl}_n(\mathbb{C})$. We prove that the preimage of $0$ under $\mu$ is a complete intersection when $\mathfrak{p}$ has finitely many $P$-orbits, where $P\subseteq \operatorname{GL}_n(\mathbb{C})$ is a parabolic subgroup such that $\operatorname{Lie}(P) = \mathfrak{p}$, and give an explicit description of the irreducible components. This allows us to study nearby fibers of $\mu$ as they are equidimensional, and one may also construct GIT quotients $\mu^{-1}(0) /\!\!/_{\chi} P$ by varying the stability condition $\chi$. Finally, we study a variety analogous to the scheme studied by Wilson with connections to a Calogero--Moser phase space where only some of particles interact.
• Double Grothendieck Polynomials and Colored Lattice Models
  Valentin Buciumas; Travis Scrimshaw
  International Mathematics Research Notices, Oxford University Press ({OUP}), 30 Dec. 2020
  English, Scientific journal, Abstract
  We construct an integrable colored six-vertex model whose partition function is a double Grothendieck polynomial. This gives an integrable systems interpretation of bumpless pipe dreams and recent results of Weigandt relating double Grothendieck polynomias with bumpless pipe dreams. For vexillary permutations, we then construct a new model that we call the semidual version model. We use our semidual model and the five-vertex model of Motegi and Sakai to give a new proof that double Grothendieck polynomials for vexillary permutations are equal to flagged factorial Grothendieck polynomials. Taking the stable limit of double Grothendieck polynomials, we obtain a new proof that the stable limit is a factorial Grothendieck polynomial as defined by McNamara. The states of our semidual model naturally correspond to families of nonintersecting lattice paths, where we can then use the Lindström–Gessel–Viennot lemma to give a determinant formula for double Schubert polynomials corresponding to vexillary permutations.
• Colored five‐vertex models and Lascoux polynomials and atoms
  Valentin Buciumas; Travis Scrimshaw; Katherine Weber
  Journal of the London Mathematical Society, 102, 3, 1047, 1066, Wiley, Dec. 2020
  English, Scientific journal
• Existence of Kirillov–Reshetikhin Crystals for Multiplicity-Free Nodes
  Rekha Biswal; Travis Scrimshaw
  Publications of the Research Institute for Mathematical Sciences, 56, 4, 761, 778, European Mathematical Society Publishing House, 07 Oct. 2020
  English, Scientific journal
• Uniform description of the rigged configuration bijection
  Travis Scrimshaw
  Selecta Mathematica, Jul. 2020
  English, Scientific journal
• Crystal structures for canonical Grothendieck functions
  Graham Hawkes; Travis Scrimshaw
  Algebraic Combinatorics, Jun. 2020
  English, Scientific journal
• Multiline Queues with Spectral Parameters
  Erik Aas; Darij Grinberg; Travis Scrimshaw
  Communications in Mathematical Physics, Mar. 2020
  English, Scientific journal
• On higher level Kirillov–Reshetikhin crystals, Demazure crystals, and related uniform models
  Cristian Lenart; Travis Scrimshaw
  Journal of Algebra, Dec. 2019
  English, Scientific journal
• Correction to: Categorical Relations Between Langlands Dual Quantum Affine Algebras: Exceptional Cases
  Se-jin Oh; Travis Scrimshaw
  Communications in Mathematical Physics, Oct. 2019
  English, Scientific journal
• Identities from representation theory
  Se-jin Oh; Travis Scrimshaw
  Discrete Mathematics, Sep. 2019
  English, Scientific journal
• Kirillov–Reshetikhin crystals $B^{1,s}$ for $\widehat{\mathfrak{sl } }_{n}$ using Nakajima monomials
  Emily Gunawan; Travis Scrimshaw
  Algebras and Representation Theory, 18 Jun. 2019
  English, Scientific journal
• A Uniform Approach to Soliton Cellular Automata Using Rigged Configurations
  Xuan Liu; Travis Scrimshaw
  Annales Henri Poincaré, Apr. 2019
  English, Scientific journal
• Categorical Relations Between Langlands Dual Quantum Affine Algebras: Exceptional Cases
  Se-jin Oh; Travis Scrimshaw
  Communications in Mathematical Physics, 28 Feb. 2019
  English, Scientific journal
• Hall-Littlewood polynomials and a Hecke action on ordered set partitions
  Jia Huang; Brendon Rhoades; Travis Scrimshaw
  Proceedings of the American Mathematical Society, 29 Jan. 2019
  Scientific journal
• Alcove path model for $B(\infty)$
  Arthur Lubovsky; Travis Scrimshaw
  Journal of Pure and Applied Algebra, 2019
  Scientific journal
• Virtualization map for the Littelmann path model
  Jianping Pan; Travis Scrimshaw
  Transformation Groups, 10 Dec. 2018
  Scientific journal
• Virtual crystals and Nakajima monomials
  Ben Salisbury; Travis Scrimshaw
  SIGMA Symmetry Integrability Geometry Methods and Applications, Oct. 2018
  Scientific journal
• Rigged configuration bijection and the proof of the X=M conjecture for nonexceptional affine types.
  Masato Okado; Anne Schilling; Travis Scrimshaw
  Journal of Algebra, 2018
  Scientific journal
• Rigged configurations and the *-involution
  Ben Salisbury; Travis Scrimshaw
  Letters in Mathematical Physics, 2018
  Scientific journal
• Type $D_n^{(1)}$ rigged configuration bijection
  Masato Okado; Reiho Sakamoto; Anne Schilling; Travis Scrimshaw
  Journal of Algebraic Combinatorics, 46, 2, 341, 401, Springer Nature, 07 Apr. 2017
  English, Scientific journal
• Rigged configurations for all symmetrizable types
  Salisbury, Ben; Scrimshaw, Travis
  Electronic Journal of Combinatorics, 24, 1, 17 Feb. 2017
  Scientific journal
• Rigged configurations as tropicalizations of loop Schur functions
  Scrimshaw, Travis
  Journal of Integrable Systems, 2, 1, Feb. 2017
  Scientific journal
• Connecting Marginally Large Tableaux and Rigged Configurations via Crystals
  Ben Salisbury; Travis Scrimshaw
  Algebras and Representation Theory, 19, 3, 523, 546, Springer Nature, 09 Feb. 2016
  Scientific journal
• A crystal to rigged configuration bijection and the filling map for type $D_4^{(3)}$
  Travis Scrimshaw
  Journal of Algebra, 2016
  Scientific journal
• Crystal structure on rigged configurations and the filling map
  Ben Salisbury; Travis Scrimshaw
  Electronic Journal of Combinatorics, 2015
  Scientific journal
• A rigged configuration model for $B(\infty)$
  Ben Salisbury; Travis Scrimshaw
  Journal of Combinatorial Theory. Series A, 2015
  Scientific journal
• Abrams's stable equivalence for graph braid groups
  Paul Prue; Travis Scrimshaw
  Topology and its Applications, 178, 136, 145, Elsevier {BV}, 2014
  Scientific journal

• 大学院共通授業科目（教育プログラム）：新渡戸カレッジオナーズプログラム大学院カリキュラム, 2024年, 修士課程, 大学院共通科目
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• Supersymmetry in the geometry of particle systems
  科学研究費助成事業
  01 Apr. 2023 - 31 Mar. 2026
  Scrimshaw Travis
  日本学術振興会, 若手研究, 北海道大学, 23K12983
• Supersymmetric rigged configurations and crystals of quantum affine supergroups
  Grants-in-Aid for Scientific Research
  18 Nov. 2021 - 31 Mar. 2024
  尾角 正人; SCRIMSHAW TRAVIS
  I completed two projects related to (skew) Howe duality with collaborators that we were working on during the past two years. The papers have appeared on the arXiv preprint server. One is a rational lifting of a classical result to help us better understand the invariants. The other develops a new probabilistic model on partitions that results in a new link with coding theory through Krawtchouk polynomials. I and collaborators have also connected other probabilistic models and Schubert calculus. We are writing multiple papers based on these results. Two students I have been mentoring obtained results on a superalgebra analog of soliton cellular automata, which has been accepted for a highly competitive talk at the international conference "Formal Power Series and Algebraic Combinatorics."
  Japan Society for the Promotion of Science, Grant-in-Aid for JSPS Fellows, 21F51028