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Algebra (Travis Scrimshaw) Lab
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Research that rewrites representation theory in the language of combinatorics and connects it to physics and probability theory.
AboutAbout the lab
Symmetry, in the language of combinatorics.
Prof. Scrimshaw studies how to express representation theory in the language of combinatorics.
Representation theory is a field that uses linear algebra to study algebraic objects and to understand the symmetries of physical systems.
Using Prof. Kashiwara's crystal theory, Prof. Scrimshaw works on representations of Kac-Moody Lie algebras and Lie superalgebras.
Prof. Scrimshaw applies this to Schubert calculus, which deals with the geometry of subspaces of vector spaces, and also explores connections with physics and probability theory.
Key terms
- Representation theory
- A field that studies algebraic objects such as groups by recasting them in the language of matrices (linear algebra).
- Crystal theory
- A theory that expresses a special basis of a quantum group representation combinatorially, as a graph of points and colored arrows.
- Schubert calculus
- A field that treats the geometry formed by the collection of subspaces of a vector space as an enumeration problem.
ResearchResearch themes
The lab studies the area where combinatorics and representation theory overlap. Many of the problems originate in mathematical physics, such as integrable systems and quantum groups.
*Illustrative imageUnderstanding representations through crystals
A crystal is a combinatorial realization of a special basis of a representation of a quantum group. When it is expressed as a directed graph with colored edges, algebraic operations such as tensor products and duals are replaced by combinatorial rules.
*Illustrative imageTaking on the X=M conjecture
The lab is working on a program to construct a correspondence between affine-type Kirillov–Reshetikhin crystals and rigged configurations, which arose from mathematical physics. This correspondence is also linked to the dynamics of soliton cellular automata, which generalize the box-ball system.
*Illustrative imageExtension to K-theory
This research revisits symmetric Grothendieck polynomials, which appear in the K-theory of Grassmannians, from the viewpoint of crystal structure. In joint work, the lab put a crystal structure on set-valued tableaux and proposed a structure called a K-crystal.
*Illustrative imageSolvable lattice models and stochastic processes
Using colored solvable lattice models, the lab proves combinatorial properties of symmetric Grothendieck polynomials. It also investigates connections with particle processes such as the TASEP on a line.
MembersFaculty
- Associate ProfessorTravis Scrimshaw
VoicesStudent voices
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Report an errorAbout the images
The top and research theme images are illustrative and were not taken at the lab. They are cropped from Wikimedia Commons images. Cropped CC BY-SA images are provided under the same CC BY-SA license as the original (version shown on each line).
- Top image: “E10 e8coxeterplane height120” Teake Nutma / CC BY 3.0 (cropped)
- Understanding representations through crystals: “A3vzome” Mathphysman / CC BY-SA 4.0 (cropped)
- Taking on the X=M conjecture: “Rule30-256-rows” Zhiming Wang / CC0 (cropped)
- Extension to K-theory: “Aztec diamond, n=50” MiaFr / CC BY-SA 3.0 (cropped)
- Solvable lattice models and stochastic processes: “Illustration of Yang Baxter Equation” AHusain314 / Public domain (cropped)
- Related lab (Analysis Lab (Takahiro Hasebe)): “Fig sinc” Sergey69 / CC BY-SA 3.0 (cropped)
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- Related lab (Kawasaki Lab): “Chaos Theory & Double Pendulum - 1” Cristian V. / CC BY-SA 4.0 (cropped)
- Related lab (Analysis (Kobayashi) Lab): “Unknown Sound Spectrogram” Daniel Hilse / CC BY-SA 4.0 (cropped)
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