Nakashima Toshiki

(中島俊樹)

Profile Information

Affiliation: Professor, Faculty of Science and Technology, Department of Information and Communication Sciences, Sophia University

Degree: 博士（理学）(京都大学)

Contact information: toshikisophia.ac.jp
Researcher number: 60243193
J-GLOBAL ID: 200901093297244007
researchmap Member ID: 1000030180

(Subject of research)
Representation theory of quantum groups--geometric crystal,crystal base

Research Interests

Research Areas

Natural sciences / Algebra / Quantum group, Representation theory, Crystal base, Geometric crystal, Cluster algebra

Education

Apr, 1988 - Mar, 1992

Kyoto University
Apr, 1984 - Mar, 1988

Faculty of Science, The University of Tokyo

Papers

D(1) 7 - Geometric Crystal at the Spin Node

Kailash C. Misra, Toshiki Nakashima, Suchada Pongpraser

Algebras and Representation Theory, 28(2) 509-530, Mar, 2025 Peer-reviewed
Geometric crystal on unipotent variety

Toshiki Nakashima

Proceedings of Symposia in Pure Mathematics, 108 465-481, Jul, 2024 Peer-reviewedInvitedLead author
An Algorithm for Berenstein–Kazhdan Decoration Functions and Trails for Classical Lie Algebras

Yuki Kanakubo, Gleb Koshevoy, Toshiki Nakashima

International Mathematics Research Notices, 2024(4) 3223-3277, Feb, 2024 Peer-reviewed
HALF POTENTIAL ON GEOMETRIC CRYSTALS AND CONNECTEDNESS OF CELLULAR CRYSTALS

YUKI KANAKUBO, TOSHIKI NAKASHIMA

Transformation Groups, 28(1) 327-373, Dec 29, 2023 Peer-reviewedLead author
An algorithm for Berenstein-Kazhdan decoration functions and trails for minuscule representations

Yuki Kanakubo, Gleb Koshevoy, Toshiki Nakashima

Journal of Algebra, 608 106-142, Oct, 2022 Peer-reviewed
Explicit forms of polyhedral realizations of crystal bases associated with adapted sequences

T.Nakashima (with, Y.Kanakubo

Toyama Mathematical Journal, 42 1-28, 2021 Peer-reviewedInvitedLast author
Adapted Sequences and Polyhedral Realizations of Crystal Bases for highest weight modules

Toshiki Nakashima, with Yuki kanakubo

Journal of Algebra, 574 327-374, 2021 Peer-reviewed
Adapted sequence for polyhedral realization of crystal bases

Yuki Kanakubo, Toshiki Nakashima

Communications in Algebra, online, Jun, 2020 Peer-reviewed
Decomposition Theorem for Product of Fundamental Crystals in Monomial Realization

Manal Alshuqayr, Toshiki Nakashima

Tokyo Journal of Mathematics, 43(1) 239-258, Jun, 2020 Peer-reviewed
Geometric crystals and Cluster ensembles in Kac-Moody setting

Yuki kanakubo, Toshiki Nakashima

Journal of Geometry and Physics, 149 103576, Mar, 2020 Peer-reviewed
Cluster algebras of finite type via Coxeter elements and Demazure crystals of type A.

Toshiki Nakashima, with, Yuki Kanakubo

Journal of Algebra, 538 149-206, 2019 Peer-reviewed
Cluster variables double Bruhat cells G{u,e} of Classical Groups and Monomial Realizations of Demazure Crystals

Y.Kanakubo, Toshiki Nakashima

International Mathematics Research Notices, 2018(12) 3621-3670, Jun, 2018 Peer-reviewed
Affine geometric crystal of A^{(1)}_n and limit of Kirillov-Reshetikhin perfect crystals

Toshiki Nakashima, withKailash C. Misra

Jouranl of Algebra, 507 249-291, Apr 27, 2018 Peer-reviewed
Explicit forms of cluster variables on double bruhat cells gu,e of type c

Yuki Kanakubo, Toshiki Nakashima

Tokyo Journal of Mathematics, 39(3) 643-678, Mar 1, 2017 Peer-reviewedInvited
Cluster Variables on Certain Double Bruhat Cells of Type (u, e) and Monomial Realizations of Crystal Bases of Type A

Yuki Kanakubo, Toshiki Nakashima

SYMMETRY INTEGRABILITY AND GEOMETRY-METHODS AND APPLICATIONS, 11(033) 32, 2015 Peer-reviewed
Decorated Geometric Crystals and Polyhedral Realizations of type D_n

Toshiki Nakashima

Contemporary Mathematics, 623 227-242, Sep, 2014 Peer-reviewedInvited
Decorations on geometric crystals and monomial realizations of crystal bases for classical groups

Toshiki Nakashima

JOURNAL OF ALGEBRA, 399(1) 712-769, Feb, 2014 Peer-reviewed
Decorated Geometric Crystals, Polyhedral and Monomial Realizations of Crystal Bases

Toshiki Nakashima

Contemporary Mathematics, 602 143-163, Dec, 2013 Peer-reviewedInvited
RSK Type Correspondence of Pictures and Littlewood-Richardson Crystals

Toshiki Nakashima, Miki Shimojo

TOKYO JOURNAL OF MATHEMATICS, 36(1) 113-130, Jun, 2013 Peer-reviewed
A(1)n -geometric crystal corresponding to Dynkin index i = 2 and its ultra-discretization

Kailash C. Misra, Toshiki Nakashima

Springer Proceedings in Mathematics and Statistics, 40 297-318, 2013 Peer-reviewed
Ultra-discretization of the D 4 (3) -geometric crystal to the G 2 (1) -perfect crystals

Mana Igarashi, Kailash C. Misra, Toshiki Nakashima

Pacific Journal of Mathematics, 255(1) 117-142, 2012 Peer-reviewed
Pictures and Littlewood-Richardson Crystals

Toshiki Nakashima, Miki Shimojo

TOKYO JOURNAL OF MATHEMATICS, 34(2) 493-506, Dec, 2011 Peer-reviewed
Geometric crystals on flag varieties and unipotent subgroups of classical groups

Maria Igarashi, Toshiki Nakashima

JOURNAL OF GEOMETRY AND PHYSICS, 61(11) 2267-2284, Nov, 2011 Peer-reviewed
ADMISSIBLE PICTURES AND LITTLEWOOD-RICHARDSON CRYSTALS

Toshiki Nakashima, Miki Shimojo

COMMUNICATIONS IN ALGEBRA, 39(10) 3849-3865, 2011 Peer-reviewed
Tropical R maps and affine geometric crystals

Masaki Kashiwara, Toshiki Nakashima, Masato Okado

Representation Theory, 14(14) 446-509, Jul 7, 2010 Peer-reviewed
Affine Geometric Crystal of type D(4)((3))

Mana Igarashi, Toshiki Nakashima

QUANTUM AFFINE ALGEBRAS, EXTENDED AFFINE LIE ALGEBRAS, AND THEIR APPLICATIONS, 506(506) 215-226, 2010 Peer-reviewed
Epsilon Systems on Geometric Crystals of type A(n)

Toshiki Nakashima

SYMMETRY INTEGRABILITY AND GEOMETRY-METHODS AND APPLICATIONS, 6(023) 14, 2010 Peer-reviewed
Ultra-Discretization of the G(2)((1))-Geometric Crystals to the D(4)((3))-Perfect Crystals

Toshiki Nakashima

REPRESENTATION THEORY OF ALGEBRAIC GROUPS AND QUANTUM GROUPS, 284 273-296, 2010 Peer-reviewed
Universal Tropical R map of $\mathfrak{sl}_2$ and Prehomogeneous Geometric Crystals (New Trends in Combinatorial Representation Theory)

NAKASHIMA TOSHIKI

RIMS Kokyuroku Bessatsu, B11 101-116, Apr, 2009 Peer-reviewed
Affine geometric crystals and limit of perfect crystals

Masaki Kashiwara, Toshiki Nakashima, Masato Okado

TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY, 360(7) 3645-3686, 2008 Peer-reviewed
Affine geometric crystal of type G(2)((1))

Toshiki Nakashima

LIE ALGEBRAS, VERTEX OPERATOR ALGEBRAS AND THEIR APPLICATIONS, 442 179-192, 2007 Peer-reviewed
Evaluation representations of quantum affine algebras at roots of unity

Yuuki Abe, Toshiki Nakashima

Journal of Mathematical Physics, 47(8) 083514-1 --- 083514-28, Sep, 2006 Peer-reviewed
Affine geometric crystals and tropical R

NAKASHIMA TOSHIKI

Proceedings of the conference on Groups and Lie Algebras, 7-23, Jan, 2006
Geometric crystals and affine crystals

Toshiki Nakashima

Representations of Algebraic Groups, Quantum Groups, and Lie Algebras, 413(413) 157-171, 2006
Geometric crystals on unipotent groups and generalized Young tableaux

Toshiki Nakashima

JOURNAL OF ALGEBRA, 293(1) 65-88, Nov, 2005 Peer-reviewed
Tropical Affine Crystals (Combinatorial Aspect of Integrable Systems)

NAKASHIMA TOSHIKI

RIMS Kokyuroku, 1429 43-56, Apr, 2005
Geometric crystals on Schubert varieties

Toshiki Nakashima

JOURNAL OF GEOMETRY AND PHYSICS, 53(2) 197-225, Feb, 2005 Peer-reviewed
Polyhedral realizations of crystal bases for modified quantum algebras of type A

A Hoshino, Toshiki Nakashima

COMMUNICATIONS IN ALGEBRA, 33(7) 2167-2191, 2005 Peer-reviewed
Nilpotent representations of classical quantum groups at roots of unity

NAKASHIMA TOSHIKI, Y.Abe

Journal of Matthematical Physics, 46(11) 113505, 2005 Peer-reviewed
Geometric Crystals and Generalized Young Tableaux (Aspects of Combinatorial Representaion Theory)

NAKASHIMA TOSHIKI

RIMS Kokyuroku, 1382 143-153, Jun, 2004
Extremal projectors of q-boson algebras

Toshiki Nakashima

COMMUNICATIONS IN MATHEMATICAL PHYSICS, 244(2) 285-296, Jan, 2004 Peer-reviewed
On certain maximal cyclic modules for the quantized special linear algebra at a root of unity

M Kaneda, Toshiki Nakashima

PACIFIC JOURNAL OF MATHEMATICS, 211(2) 273-282, Oct, 2003 Peer-reviewed
Geometric Crystals, Schubert Varieties and Crystal Bases (Combinatorial Representation Theory and Related Topics)

NAKASHIMA TOSHIKI

RIMS Kokyuroku, 1310 186-201, Apr, 2003
Irreducible modules of finite dimensional quantum algebras of type A at roots of unity

Toshiki Nakashima

JOURNAL OF MATHEMATICAL PHYSICS, 43(4) 2000-2014, Apr, 2002 Peer-reviewed
Polytopes for crystallized demazure modules and extremal vectors

Toshiki Nakashima

COMMUNICATIONS IN ALGEBRA, 30(3) 1349-1367, 2002 Peer-reviewed
１の巾根における有限次元A型量子群の既約表現について

中島俊樹

代数群と量子群の表現論研究集会報告集, 4, 2001
Polyhedral realizations of crystal bases for integrable highest weight modules

Toshiki Nakashima

JOURNAL OF ALGEBRA, 219(2) 571-597, Sep, 1999 Peer-reviewed
結晶基底の多面体表示について

中島俊樹

代数群と量子群の表現論研究集会報告集, 2, 1999
Polyhedral Realizations of Crystal Bases and Braid-type Isomorphisms

NAKASHIMA TOSHIKI

Contemporary Mathematics, 248 419-435, 1999 Peer-reviewed
Polyhedral realizations of crystal bases for quantized Kac-Moody algebras

Toshiki Nakashima, A Zelevinsky

ADVANCES IN MATHEMATICS, 131(1) 253-278, Oct, 1997 Peer-reviewed

Misc.

A survey on categorified crystal structure on localized quantum coordinate rings

Toshiki Nakashima

RIMS Kôkyûroku, (2258) 20-36, Jun, 2023 InvitedLead author
CRYSTAL STRUCTURE ON LOCALIZED QUANTUM COORDINATE RINGS

Toshiki Nakashima

RIMS Kôkyûroku, (2234), Nov, 2022 InvitedLead author
団代数と結晶基底 (特集団代数をめぐって : 新たな共通構造の認識)

中島俊樹

数理科学, 53(3) 21-26, Mar, 2015
幾何クリスタルと可積分系(可積分系数理の眺望)

柏原正樹, 中島俊樹, 尾角正人

数理解析研究所講究録, 1541 150-165, Apr, 2007
書評 J. Hong and S.-J. Kang: Introduction to Quantum Groups and Crystal Bases (Grad. Stud. Math., 42)

中島俊樹

数学, 58(1) 102-111, Jan, 2006 Peer-reviewedInvited

Books and Other Publications

数理科学3月号特集「量子群の世界」, 量子群入門

中島俊樹 (Role: Contributor, 量子群入門)

サイエンス社, Mar, 2025
数理科学3月号特集「量子群の世界」, 巻頭言

中島俊樹 (Role: Contributor, 巻頭言)

サイエンス社, Mar, 2025
『結晶基底と幾何結晶』臨時別冊・数理科学SGCライブラリ148

中島俊樹

サイエンス社, Apr 25, 2019
数理科学、団代数と結晶基底 (特集団代数をめぐって : 新たな共通構造の認識)

中島俊樹

サイエンス社, Mar 1, 2015
数学セミナー、量子群と結晶基底 (特集図形から学ぶリー代数)

中島俊樹

日本評論社, Oct 1, 2009

Presentations

Crystal structure of localized quantum unipotent coordinate category,

Toshiki Nakashima

Combinatorics and Arithmetics for physics, Nov 21, 2024 Invited
Crystal structure of localized quantum unipotent coordinate category

Toshiki Nakashima

Quantum Gropus and representation theory,, Oct 13, 2024 Invited
Presentation of Crystal bases

Toshiki Nakashima

Crystal Bases and Then ..., Jul 11, 2024 Invited
Localized Quantum Unipotent Coordinate Category and Cellular Crystals

Toshiki Nakashima

Combinatorial Representation Theory and Geometry, Jun 26, 2024 Invited
Categorified Crystal Bases on Localized Quantum Coordinate Rings and Cellular Crystals

Toshiki Nakashima

Combinatorics and Arithmetic for Physics, Nov 16, 2023 Invited

Professional Memberships

日本数学会

Research Projects

Development from crystal bases to categorification, cluster algebra and geometric crystal

Apr, 2025 - Mar, 2029

Toshiki Nakashima
幾何結晶とクラスター代数への結晶基底からのアプローチ

Apr, 2020 - Mar, 2025
クラスター代数と結晶基底の表現論的研究

Apr, 2015 - Mar, 2020

中島俊樹
Representation theory of Iwahori-Hecke algebras and reflection groups

Grants-in-Aid for Scientific Research, Japan Society for the Promotion of Science, Apr, 2011 - Mar, 2016

Gomi Yasushi, NAKASHIMA Toshiki, SHINODA Ken-ichi, Ma. Louise Antonette N. De Las Penas, Loyola Mark L.
Algebraic analysis of infinite symmetry

Grants-in-Aid for Scientific Research, Japan Society for the Promotion of Science, Apr, 2010 - Mar, 2015

KASHIWARA Masaki, ARIKI Susumu, TANISAKI Toshiyuki, NAKASHIMA Toshiki, KATO Syu, MIWA Tetsuji, SCHAPIRA Pierre, KANG Seok-Jin, VILONEN Kari, D'AGNOLO Andrea

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