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Hikita Lab
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The lab is a geometric representation theory group that explores duality, using problems in combinatorics as a guide.
AboutAbout the lab
Finding dualities, with combinatorics as a probe
The Hikita Lab studies geometric representation theory.
In mathematics and mathematical physics, a phenomenon called “duality” has been observed, in which seemingly completely different mathematical objects turn out to be equivalent. Mirror symmetry and Langlands duality are well-known examples.
One of the lab's research goals is to use problems in combinatorics as probes to find phenomena like duality that lie behind them.
The lab has also proved the Stanley–Stembridge conjecture on chromatic symmetric functions of graphs.
Key terms
- Duality
- A phenomenon in which seemingly completely different mathematical objects turn out to be equivalent.
- Combinatorics
- A branch of mathematics that studies discrete structures, such as ways of arranging and counting objects, and graphs.
- Representation theory
- A field that studies abstract structures such as groups and algebras by representing them concretely as matrices and linear transformations.
NewsLab news
View More ↗ResearchResearch themes
One of the lab's goals is to find dualities, phenomena in which seemingly completely different mathematical objects are equivalent, using problems in combinatorics, a hypothetical field of application, as probes. Computing the same thing by different mathematical methods can sometimes yield more concrete, nontrivial statements.
*Illustrative imageTopological interpretation of the shuffle conjecture
Starting from the shuffle conjecture, an equality between two symmetric functions, the lab gave a topological interpretation using the homology of affine Springer fibers of type A.
*Illustrative imageSymplectic duality
Under symplectic duality, the lab discovered a phenomenon in which a topological object, the cohomology ring, and an algebro-geometric object, a certain quotient of the coordinate ring, are interchanged.
*Illustrative imageCanonical bases and highest weight categories
To deepen understanding on the algebraic geometry side, the lab formulated a conjecture that the derived category of equivariant coherent sheaves on a symplectic resolution can be equipped with a representation-theoretic structure (a highest weight category).
*Illustrative imageElliptic canonical bases
As a refinement of the theory of canonical bases in another direction, the lab discusses lifts to elliptic cohomology in the case of toric hyperkähler varieties.
MembersFaculty
- Associate ProfessorTatsuyuki Hikita
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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: “Young tableau for partitions of number 10 stacked” Unmismoobjetivo / CC BY 4.0 (cropped)
- Topological interpretation of the shuffle conjecture: “Elementary symmetric functions (1)” Daviddaved / CC BY 3.0 (cropped)
- Symplectic duality: “Weyl chambers for A2” Mathphysman / CC BY-SA 4.0 (cropped)
- Canonical bases and highest weight categories: “Young-Schensted” ManiacParisien / CC BY-SA 3.0 (cropped)
- Elliptic canonical bases: “The lemniscate sine and cosine related to the arclength of the lemniscate of Bernoulli” Jacob Rus / CC BY-SA 4.0 (cropped)
- Related lab (Applied Mathematics Lab (Fukuhiro Ueda)): “Eisenstein 14” Jan Homann / Public domain (cropped)
- Related lab (Applied Mathematics Division (Kawakita) Lab): “Du Val singularities” Aizenr / CC BY-SA 4.0 (cropped)
- Related lab (Applied Mathematics Division (Kawamura) Lab): “Turing Machine” Porao / Public domain (cropped)
- Related lab (Applied Mathematics Lab (Kazuhisa Makino)): “B functies 2var versie02” Bob.v.R / CC BY-SA 4.0 (cropped)
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