Updated on 2025/07/01

写真a

 
KOUNO, Takafumi
 
Affiliation
Faculty of Science and Engineering, School of Fundamental Science and Engineering
Job title
Assistant Professor(non-tenure-track)
Degree
Doctor of Science ( 2021.09 Tokyo Institute of Technology )

Research Experience

  • 2024.04
    -
    Now

    Waseda University   Faculty of Science and Engineering   Assistant Professor

  • 2023.10
    -
    2024.03

    Waseda University   Research Institute for Science and Engineering   Junior Researcher

  • 2022.04
    -
    2024.03

    Waseda University   Faculty of Science and Engineering   JSPS Research Fellowship for Young Scientists PD

  • 2021.10
    -
    2022.03

    Tokyo Institute of Technology   School of Science   JSPS Research Fellowship for Young Scientists PD (changed from DC2)

  • 2020.04
    -
    2021.09

    Tokyo Institute of Technology   School of Science   JSPS Research Fellowship for Young Scientists DC2

Education Background

  • 2018.04
    -
    2021.09

    Tokyo Institute of Technology   School of Science   Doctoral course, Department of Mathematics  

  • 2016.04
    -
    2018.03

    Tokyo Institute of Technology   School of Science   Master's course, Department of Mathematics  

  • 2012.04
    -
    2016.03

    Tokyo Institute of Technology   School of Science   Department of Mathematics  

Professional Memberships

  • 2022.04
    -
    Now

    The Mathematical Society of Japan

Research Areas

  • Algebra / Geometry

Research Interests

  • アフィン量子群

  • 旗多様体

  • 量子K環

  • 量子alcoveモデル

  • Chevalley公式

 

Papers

  • A Generalization of Quantum Lakshmibai-Seshadri Paths for an Arbitrary Weight

    Takafumi Kouno, Satoshi Naito

    Algebras and Representation Theory   27 ( 6 ) 2321 - 2353  2024.12

     View Summary

    Abstract

    We construct an injective weight-preserving map (called the forgetful map) from the set of all admissible subsets in the quantum alcove model associated to an arbitrary weight. The image of this forgetful map can be explicitly described by introducing the notion of “interpolated quantum Lakshmibai-Seshadri (QLS for short) paths”, which can be thought of as a generalization of quantum Lakshmibai-Seshadri paths. As an application, we reformulate, in terms of interpolated QLS paths, an identity of Chevalley type for the graded characters of Demazure submodules of a level-zero extremal weight module over a quantum affine algebra, which is a representation-theoretic analog of the Chevalley formula for the torus-equivariant K-group of a semi-infinite flag manifold.

    DOI

    Scopus

  • Quantum K-theory Chevalley formulas in the parabolic case

    Takafumi Kouno, Cristian Lenart, Satoshi Naito, Daisuke Sagaki

    Journal of Algebra   645   1 - 53  2024.05

    DOI

    Scopus

    2
    Citation
    (Scopus)
  • New structure on the quantum alcove model with applications to representation theory and Schubert calculus

    Takafumi Kouno, Cristian Lenart, Satoshi Naito

    Journal of Combinatorial Algebra   7 ( 3 ) 347 - 400  2023.10

     View Summary

    The quantum alcove model associated to a dominant weight plays an important role in many branches of mathematics, such as combinatorial representation theory, the theory of Macdonald polynomials, and Schubert calculus. For a dominant weight, it is proved by Lenart–Lubovsky that the quantum alcove model does not depend on the choice of a reduced alcove path, which is a shortest path of alcoves from the fundamental one to its translation by the given dominant weight. This is established through quantum Yang–Baxter moves, which biject the objects of the models associated to two such alcove paths, and can be viewed as a generalization of jeu de taquin slides to arbitrary root systems. The purpose of this paper is to give a generalization of quantum Yang–Baxter moves to the quantum alcove model corresponding to an arbitrary weight, which was used to express a general Chevalley formula for the equivariant K -group of semi-infinite flag manifolds. The generalized quantum Yang–Baxter moves give rise to a “sijection” (bijection between signed sets), and are shown to preserve certain important statistics, including weights and heights. As an application, we prove that the generating function of these statistics does not depend on the choice of a reduced alcove path. Also, we obtain an identity for the graded characters of Demazure submodules of level-zero extremal weight modules over a quantum affine algebra, which can be thought of as a representation-theoretic analogue of the mentioned Chevalley formula.

    DOI

  • Identities of Inverse Chevalley Type for the Graded Characters of Level-Zero Demazure Submodules over Quantum Affine Algebras of Type C

    Takafumi Kouno, Satoshi Naito, Daniel Orr

    Algebras and Representation Theory   27 ( 1 ) 429 - 460  2023.08

     View Summary

    Abstract

    We provide identities of inverse Chevalley type for the graded characters of level-zero Demazure submodules of extremal weight modules over a quantum affine algebra of type C. These identities express the product $$e^{\mu } \text {gch} ~V_{x}^{-}(\lambda )$$ of the (one-dimensional) character $$e^{\mu }$$, where $$\mu $$ is a (not necessarily dominant) minuscule weight, with the graded character gch$$V_{x}^{-}(\lambda )$$ of the level-zero Demazure submodule $$V_{x}^{-}(\lambda )$$ over the quantum affine algebra $$U_{\textsf{q } }(\mathfrak {g}_{\textrm{af } })$$ as an explicit finite linear combination of the graded characters of level-zero Demazure submodules. These identities immediately imply the corresponding inverse Chevalley formulas for the torus-equivariant K-group of the semi-infinite flag manifold $$\textbf{Q}_{G}$$ associated to a connected, simply-connected and simple algebraic group G of type C. Also, we derive cancellation-free identities from the identities above of inverse Chevalley type in the case that $$\mu $$ is a standard basis element $${\varepsilon }_{k}$$ in the weight lattice P of G.

    DOI

    Scopus

  • Chevalley formula for anti-dominant minuscule fundamental weights in the equivariant quantum K-group of partial flag manifolds

    Takafumi Kouno, Satoshi Naito, Daisuke Sagaki

    Journal of Combinatorial Theory, Series A   192  2022.11

    DOI

    Scopus

    5
    Citation
    (Scopus)
  • InverseK-Chevalley formulas for semi-infinite flag manifolds, I: minuscule weights in ADE type

    Takafumi Kouno, Satoshi Naito, Daniel Orr, Daisuke Sagaki

    Forum of Mathematics, Sigma   9  2021.07

     View Summary

    Abstract

    We prove an explicit inverse Chevalley formula in the equivariantK-theory of semi-infinite flag manifolds of simply laced type. By an ‘inverse Chevalley formula’ we mean a formula for the product of an equivariant scalar with a Schubert class, expressed as a$\mathbb {Z}\left [q^{\pm 1}\right ]$-linear combination of Schubert classes twisted by equivariant line bundles. Our formula applies to arbitrary Schubert classes in semi-infinite flag manifolds of simply laced type and equivariant scalars$e^{\lambda }$, where$\lambda $is an arbitrary minuscule weight. By a result of Stembridge, our formula completely determines the inverse Chevalley formula for arbitrary weights in simply laced type except for type$E_8$. The combinatorics of our formula is governed by the quantum Bruhat graph, and the proof is based on a limit from the double affine Hecke algebra. Thus our formula also provides an explicit determination of all nonsymmetricq-Toda operators for minuscule weights in ADE type.

    DOI

    Scopus

    4
    Citation
    (Scopus)
  • Decomposition of tensor products of Demazure crystals

    Takafumi Kouno

    Journal of Algebra   546   641 - 678  2020.03

    DOI

    Scopus

    4
    Citation
    (Scopus)

▼display all

Presentations

  • Special classes of the equivariant quantum K-ring of the flag manifold in type C

    河野 隆史

    Algebraic Lie Theory and Representation Theory 2025 

    Presentation date: 2025.05

    Event date:
    2025.05
     
     
  • Borel-type presentation of the torus-equivariant quantum K-ring of the flag manifold in type C and its application

    Takafumi Kouno  [Invited]

    MIST 2025 Workshop (Schubert calculus and related topics) 

    Presentation date: 2025.03

    Event date:
    2025.03
     
     
  • The Borel-type presentation of the equivariant quantum and classical K-theory of the flag manifold in type C

    河野 隆史  [Invited]

    第69回代数学シンポジウム 

    Presentation date: 2024.08

    Event date:
    2024.08
     
     
  • Positivity conjecture for the equivariant quantum K-ring of some partial flag manifolds

    河野 隆史

    Algebraic Lie Theory and Representation Theory 2024 

    Presentation date: 2024.05

    Event date:
    2024.05
     
     
  • C型旗多様体の同変量子K環のBorel表示

    河野 隆史, 内藤 聡

    日本数学会2024年度年会 

    Presentation date: 2024.03

    Event date:
    2024.03
     
     
  • Parabolic K-Peterson isomorphism for the Lagrangian Grassmannian

    河野 隆史

    表現論の組合せ論的側面とその周辺 

    Presentation date: 2023.10

    Event date:
    2023.10
     
     
  • Presentation of the torus-equivariant quantum K-ring of flag manifolds in type C

    河野 隆史

    Algebraic Lie Theory and Representation Theory 2023 

    Presentation date: 2023.05

    Event date:
    2023.05
     
     
  • Grassmann多様体の量子K理論におけるChevalley公式

    河野 隆史  [Invited]

    東工大表現論セミナー 

    Presentation date: 2022.12

    Event date:
    2022.12
     
     
  • Chevalley formula in the equivariant quantum K-theory of partial flag manifolds

    河野 隆史

    RIMS共同研究「組合せ論的表現論における最近の展開」 

    Presentation date: 2022.11

    Event date:
    2022.11
     
     
  • Generalized quantum Yang-Baxter moves and its application to Schubert calculus

    Takafumi Kouno, Cristian Lenart, Satoshi Naito

    34th International Conference on Formal Power Series & Algebraic Combinatorics 

    Presentation date: 2022.07

    Event date:
    2022.07
     
     
  • Generalized quantum Yang-Baxter moves

    Takafumi Kouno  [Invited]

    Conference on Algebraic Representation Theory 2021 

    Presentation date: 2021.11

    Event date:
    2021.11
     
     
  • 一般化された量子Yang-Baxter move

    河野 隆史

    2021年度表現論シンポジウム 

    Presentation date: 2021.11

    Event date:
    2021.11
     
     
  • Inverse K-Chevalley formula for type A semi-infinite flag manifolds

    河野 隆史  [Invited]

    南大阪代数セミナー 

    Presentation date: 2020.10

    Event date:
    2020.10
     
     
  • Inverse K-Chevalley formula for type A semi-infinite flag manifolds

    河野 隆史

    RIMS共同研究「組合せ論的表現論の最近の進展」 

    Presentation date: 2020.10

    Event date:
    2020.10
     
     
  • A generalization of Lakshmibai-Seshadri paths and Chevalley formula for arbitrary weights

    河野 隆史

    第3回数理新人セミナー 

    Presentation date: 2020.02

    Event date:
    2020.02
     
     
  • A generalization of Lakshmibai-Seshadri paths and Chevalley formula for arbitrary weights

    河野 隆史

    RIMS共同研究「表現論とその組合せ論的側面」 

    Presentation date: 2019.10

    Event date:
    2019.10
     
     
  • Decomposition of tensor products of Demazure crystals

    河野 隆史

    Algebraic Lie Theory and Representation Theory 2018 

    Presentation date: 2018.05

    Event date:
    2018.05
     
     
  • Decomposition of tensor products of Demazure crystals

    河野 隆史

    第23回代数学若手研究会 

    Presentation date: 2018.03

    Event date:
    2018.03
     
     
  • Decomposition of tensor product of Demazure crystals

    河野 隆史

    Algebraic Lie Theory and Representation Theory 2017 

    Presentation date: 2017.06

    Event date:
    2017.06
     
     

▼display all

Research Projects

  • 一般旗多様体に対するK-Peterson同型

    日本学術振興会  科学研究費助成事業

    Project Year :

    2024.07
    -
    2026.03
     

    河野 隆史

  • 半無限旗多様体を用いた量子Schubert calculusの研究

    日本学術振興会  科学研究費助成事業

    Project Year :

    2023.03
    -
    2024.07
     

    河野 隆史

     View Summary

    本研究では,一般旗多様体の同変量子K環の代数的構造を調べた.Buch-Chaput-Mihalcea-Perrinの結果により,同変量子K環の代数的構造は,Chevalley公式により決定される.そのため,Chevalley公式の組合せ論的な記述を目指した.研究計画では,Lenart-内藤-佐垣による旗多様体に対するChevalley公式を,加藤によるよい全射で一般旗多様体の同変量子K環へ写し,得られた一般旗多様体に対するChevalley公式のうち余分な項を打ち消すという流れを想定した.しかし,上述の打ち消しを記述するためには,量子Bruhatグラフを用いた複雑な場合分けが必要であり,困難であると判明した.
    そこで,新たにアフィンGrassmann多様体を利用することを試みた.加藤の結果により,アフィンGrassmann多様体の同変Kホモロジー環から一般旗多様体の同変量子K環へ,適切な局所化のもとで全射が存在する.アフィンGrassmann多様体の同変Kホモロジー環の記述ではYoung図形を利用でき,量子Bruhatグラフより簡潔である.そこで,この全射を用いて,アフィンGrassmann多様体の同変Kホモロジー環においてChevalley公式の打ち消しを記述することを目指した.
    2022年度は,C型のLagrangian Grassmann多様体に対して,Young図形を用いて打ち消しを記述し,既存の量子Bruhatグラフによる記述と同値であることを確かめた.また,この打ち消しが上述の全射の核を表すために十分な関係式であることを,大部分確かめた.
    並行して,C型の旗多様体の同変量子K環を,Laurent多項式環の剰余環として表示することを試み,その証明の大筋を得た.この表示も,最終目標である一般旗多様体の同変量子K環の代数的構造の決定に役立つと考えられる.

  • alcove walkおよび量子LSパスを用いたSchubert計算の研究

    日本学術振興会  科学研究費助成事業

    Project Year :

    2020.04
    -
    2022.03
     

    河野 隆史

     View Summary

    本年度は,まず昨年度発見した量子Yang-Baxter moveに関する論文を執筆した.この論文をプレプリントサーバーのarXivで公開した.また,研究集会「2021年度表現論シンポジウム」および「Conference on Algebraic Representation Theory 2021」にて,本研究の内容を講演した.
    続いて,C型の半無限旗多様体のトーラス同変K群において,指標のウェイトがminusculeウェイトである場合の逆Chevalley公式の記述の研究を行った.その結果,逆Chevalley公式の量子alcoveモデルを用いた明示的な記述を得た.この結果は,展開公式の有限性を含む.すなわち,展開公式における和が有限和であることと,展開係数が(Laurent)多項式であることを示している.一方で,この展開公式は一般に打ち消し合う項を含んでいる.これについて,特にウェイトがウェイト格子の基本ベクトルであるときは,この打ち消しを明示的に記述し,cancellation-freeな展開公式を得た.これらの結果については,現在論文を執筆中である.また,ウェイトが基本ベクトルの(-1)倍のときの打ち消しの研究については,現在進行中である.
    その他,関連する研究として,一般旗多様体の量子K群におけるChevalley公式の記述を研究した.一般旗多様体の量子K群におけるChevalley公式は,原理的には旗多様体のChevalley公式から直接得ることができるが,この展開公式は打ち消し合う項を含む.本研究では,とくに一般旗多様体がA型の2ステップ旗多様体の場合に,この打ち消しを研究した.その結果,所望のcancellation-freeなChevalley公式を記述することができた.この結果について,論文にまとめ,arXiv上で公開した.

 

Syllabus

Teaching Experience

  • Mathematics B2 Kikan (5)

    Waseda University  

    2025.04
    -
    Now
     

  • Mathematics B2 Shigen

    Waseda University  

    2024.04
    -
    2025.03
     

  • Science and Engineering Laboratory 1A

    Waseda University  

    2024.04
    -
    2024.09
     

  • Fundamental Mathematics Kikan(6)-I

    Waseda University  

    2024.04
    -
    2024.09
     

  • Linear Algebra 2

    Rikkyo University  

    2023.09
    -
    2024.03
     

  • Linear Algebra 1

    Rikkyo University  

    2023.04
    -
    2023.09
     

  • Linear Algebra 2

    Rikkyo University  

    2022.10
    -
    2023.03
     

  • Linear Algebra 1

    Rikkyo University  

    2022.04
    -
    2022.09
     

▼display all