Researcher Profile and Settings

Master

Affiliation (Master)

• Faculty of Science　Mathematics　Mathematics

Affiliation (Master)

• Faculty of Science　Mathematics　Mathematics

researchmap

Profile and Settings

Affiliation

• Hokkaido University, Department of Mathematics, Associate Professor

Degree

• PhD (Mathematics)（2015/06 University of California, Davis）

Profile and Settings

• Contact Point

  tcscrimsgmail.com

• Name (Japanese)

  Scrimshaw

• Name (Kana)

  Travis

• Name

  202201002897414434

Alternate Names

https://tscrim.github.io/

Affiliation

• Hokkaido University, Department of Mathematics, Associate Professor

Achievement

Research Areas

• Natural sciences / Algebra / Combinatorial representation theory

Published Papers

• Skew Howe duality and limit shapes of Young diagrams

  Anton Nazarov, Olga Postnova, Travis Scrimshaw

  Journal of the London Mathematical Society 0024-6107 2024/01 

  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 0025-5874 2024/01
• K-theoretic crystals for set-valued tableaux of rectangular shapes

  Oliver Pechenik, Travis Scrimshaw

  Algebraic Combinatorics 5 (3) 515 - 536 2589-5486 2022/07/07
• Candidate for the crystal B(−∞) for the queer Lie superalgebra

  Ben Salisbury, Travis Scrimshaw

  Kyoto Journal of Mathematics 62 (2) 2156-2261 2022/06/01
• Kirillov–Reshetikhin crystals $B^{7,s}$ for type $E_7^{(1)}$

  Rekha Biswal, Travis Scrimshaw

  Communications in Algebra 1 - 16 0092-7872 2022/02/08
• Quasi-solvable lattice models for and Demazure atoms and characters

  Valentin Buciumas, Travis Scrimshaw

  Forum of Mathematics, Sigma 10 2050-5094 2022 

  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 1083-4362 2021/09
• Rigged configurations and the ⁎-involution for generalized Kac–Moody algebras

  B. Salisbury, T. Scrimshaw

  Journal of Algebra 573 148 - 168 0021-8693 2021/05
• 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 0022-4049 2021/05
• The regularity of almost-commuting partial Grothendieck--Springer resolutions and parabolic analogs of Calogero--Moser varieties

  Travis Scrimshaw

  Journal of Lie Theory 2021 

  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 1073-7928 2020/12/30 

  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 0024-6107 2020/12
• 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 0034-5318 2020/10/07
• Uniform description of the rigged configuration bijection

  Travis Scrimshaw

  Selecta Mathematica 1022-1824 2020/07
• Crystal structures for canonical Grothendieck functions

  Travis Scrimshaw

  Algebraic Combinatorics 2589-5486 2020/06
• Multiline Queues with Spectral Parameters

  Travis Scrimshaw

  Communications in Mathematical Physics 0010-3616 2020/03
• On higher level Kirillov–Reshetikhin crystals, Demazure crystals, and related uniform models

  Travis Scrimshaw

  Journal of Algebra 0021-8693 2019/12
• Correction to: Categorical Relations Between Langlands Dual Quantum Affine Algebras: Exceptional Cases

  Travis Scrimshaw

  Communications in Mathematical Physics 0010-3616 2019/10
• Identities from representation theory

  Travis Scrimshaw

  Discrete Mathematics 0012-365X 2019/09
• Kirillov–Reshetikhin crystals $B^{1,s}$ for $\widehat{\mathfrak{sl } }_{n}$ using Nakajima monomials

  Travis Scrimshaw

  Algebras and Representation Theory 1386-923X 2019/06/18
• A Uniform Approach to Soliton Cellular Automata Using Rigged Configurations

  Travis Scrimshaw

  Annales Henri Poincaré 1424-0637 2019/04
• Categorical Relations Between Langlands Dual Quantum Affine Algebras: Exceptional Cases

  Travis Scrimshaw

  Communications in Mathematical Physics 0010-3616 2019/02/28
• Hall-Littlewood polynomials and a Hecke action on ordered set partitions

  Travis Scrimshaw

  Proceedings of the American Mathematical Society 2019/01/29
• Alcove path model for $B(\infty)$

  Travis Scrimshaw

  Journal of Pure and Applied Algebra 2019
• Virtualization map for the Littelmann path model

  Travis Scrimshaw

  Transformation Groups 2018/12/10
• Virtual crystals and Nakajima monomials

  Travis Scrimshaw

  SIGMA Symmetry Integrability Geometry Methods and Applications 2018/10
• Rigged configuration bijection and the proof of the X=M conjecture for nonexceptional affine types.

  Travis Scrimshaw

  Journal of Algebra 2018
• Rigged configurations and the *-involution

  Travis Scrimshaw

  Letters in Mathematical Physics 2018
• Type $D_n^{(1)}$ rigged configuration bijection

  Masato Okado, Reiho Sakamoto, Anne Schilling, Travis Scrimshaw

  Journal of Algebraic Combinatorics 46 (2) 341 - 401 0925-9899 2017/04/07
• Rigged configurations for all symmetrizable types

  Salisbury, Ben, Scrimshaw, Travis

  Electronic Journal of Combinatorics 24 (1) 2017/02/17
• Rigged configurations as tropicalizations of loop Schur functions

  Scrimshaw, Travis

  Journal of Integrable Systems 2 (1) 2017/02
• Connecting Marginally Large Tableaux and Rigged Configurations via Crystals

  Ben Salisbury, Travis Scrimshaw

  Algebras and Representation Theory 19 (3) 523 - 546 2016/02/09
• A crystal to rigged configuration bijection and the filling map for type $D_4^{(3)}$

  Travis Scrimshaw

  Journal of Algebra 2016
• Crystal structure on rigged configurations and the filling map

  Travis Scrimshaw

  Electronic Journal of Combinatorics 2015
• A rigged configuration model for $B(\infty)$

  Travis Scrimshaw

  Journal of Combinatorial Theory. Series A 2015
• Abrams's stable equivalence for graph braid groups

  Paul Prue, Travis Scrimshaw

  Topology and its Applications 178 136 - 145 2014

Research Projects

• Supersymmetry in the geometry of particle systems

  日本学術振興会：科学研究費助成事業

  Date (from‐to) : 2023/04 -2026/03 

  Author : Scrimshaw Travis
• Supersymmetric rigged configurations and crystals of quantum affine supergroups

  Japan Society for the Promotion of Science：Grants-in-Aid for Scientific Research

  Date (from‐to) : 2021/11 -2024/03 

  Author : 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."