2020 Fiscal Year Research-status Report
Combinatorics around Painleve VI
| Project/Area Number |
16K05057
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| Research Institution | Kyoto University |
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Principal Investigator |
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| Project Period (FY) |
2016-04-01 – 2022-03-31
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| Keywords | FK algebras / Classical Schubert / Quantum Schubert / Grothendieck Calculi / Classical RSK / Tropical RSK / Probabilistic models / Integrable Systems |
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| Outline of Annual Research Achievements |
I continued the study of some noncommutative algebras (the so-called Fomin--Kirillov algebras (FK algebras for short) with applications to
(A) algebraic and combinatorial properties of the Hessenberg and parabolic flag varieties, as well as their cohomological,K-theoretic properties, including Schubert and Grothendieck Calculi (with Maeno, H. Naruse)
(B) elliptic representations of FK algebras and Ruijsenaar-Macdonald elliptic operators (with M.Noumi)
(C) algebraic aspects of the Bethe Ansatz with applications to RSK and RC correspondences to combinatorics of probabilistic models (with R. Sakamoto and N. Zygouras);
(D) P-positivity and log-concavity in Matroid Theory.
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| Current Status of Research Progress |
Current Status of Research Progress
1: Research has progressed more than it was originally planned.
Reason
It is really surprising/mysterious that FK algebras fall on the crossroad of many branches of Mathematics such Algebraic Geometry and Combinatorics,Special Functions (including elliptic ones), Integrable System: classical, discrete and quantum; Low dimensional Topology (invariants of knots); Integrable Probability : some KPZ type models, and so on and so far.
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| Strategy for Future Research Activity |
I'm planning to continue my research concerning properties and applications
of (generalized) FK algebras to description of polynomials appearing in Matroid Theory, Classical and Quantum Schubert and Grothendieck Catalan, as well as in theory of Ruijsenaar-Macdonald elliptic operators; as well to study further
cluster nature of basic construction in Tropical Combinatorics.
Part of results in these direction collected in my (still in progress) book
" Lectures on Quadratic Algebras"; in my joint with R. Sakamoto book " Combinatorial Bethe Ansatz", in Japanese), 2020), Morikita Publishing Co., and still in progress, the extended English version of our book.
Also I'm planning to present some results and Conjectures on coming in October 2021 (online) Workshop "P-positivity in Matroid Theory and related Topics".
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| Causes of Carryover |
My research of algebra-geometric and combinatorisl properties of FK algebras, Schubert, Key and Grotendieck, as well as Catalan, Motzkin, poly-Bernoulli, and related ones,touches a wide variety of fields in Mathematics, such as Algebra, Algebraic Geometry and Combinatorics, Special Functions and Polynomials,
Integrable Systems, Low Dimensional Topology, Graph and Matroid theories,Integrable Probability, and several others.
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