Monomial realizations and LS paths of fundamental representations for rank 2 Kac-Moody algebras
Yuki Kanakubo, Elsevier BV
Journal of Algebra, Jun. 2025, [Reviewed]

An Algorithm for Berenstein–Kazhdan Decoration Functions and Trails for Classical Lie Algebras
Yuki Kanakubo; Gleb Koshevoy; Toshiki Nakashima, Abstract

For a simply connected connected simple algebraic group $G$, it is known that a variety $B_{w_0}^-:=B^-\cap U\overline{w_0}U$ has a geometric crystal structure with a positive structure $\theta ^-_{\textbf{i } }:(\mathbb{C}^{\times })^{l(w_0)}\rightarrow B_{w_0}^-$ for each reduced word $\textbf{i}$ of the longest element $w_0$ of Weyl group. A rational function $\Phi ^h_{BK}=\sum _{i\in I}\Delta _{w_0\Lambda _i,s_i\Lambda _i}$ on $B_{w_0}^-$ is called a half-potential, where $\Delta _{w_0\Lambda _i,s_i\Lambda _i}$ is a generalized minor. Computing $\Phi ^h_{BK}\circ \theta ^-_{\textbf{i } }$ explicitly, we get an explicit form of string cone or polyhedral realization of $B(\infty )$ for the finite dimensional simple Lie algebra $\mathfrak{g}=\textrm{Lie}(G)$. In this paper, for an arbitrary reduced word $\textbf{i}$, we give an algorithm to compute the summand $\Delta _{w_0\Lambda _i,s_i\Lambda _i}\circ \theta ^-_{\textbf{i } }$ of $\Phi ^h_{BK}\circ \theta ^-_{\textbf{i } }$ in the case $i\in I$ satisfies that for any weight $\mu $ of $V(-w_0\Lambda _i)$ and $t\in I$, it holds $\langle h_t,\mu \rangle \in \{2,1,0,-1,-2\}$. In particular, if $\mathfrak{g}$ is of type $\textrm{A}_n$, $\textrm{B}_n$, $\textrm{C}_n$ or $\textrm{D}_n$ then all $i\in I$ satisfy this condition so that one can completely calculate $\Phi ^h_{BK}\circ \theta ^-_{\textbf{i } }$. We will also prove that our algorithm works in the case $\mathfrak{g}$ is of type $\textrm{G}_2$., Oxford University Press (OUP)
International Mathematics Research Notices, Feb. 2024, [Reviewed]

Polyhedral Realizations for $B(\infty )$ and Extended Young Diagrams, Young Walls of Type $\mathrm {A}^{(1)}_{n-1}$, $\mathrm {C}^{(1)}_{n-1}$, $\mathrm {A}^{(2)}_{2n-2}$, $\mathrm {D}^{(2)}_{n}$
Yuki Kanakubo, Springer Science and Business Media LLC
Algebras and Representation Theory, Oct. 2023, [Reviewed]

Polyhedral realizations for crystal bases of integrable highest weight modules and combinatorial objects of type $\mathrm {A}^{(1)}_{n-1}$, $\mathrm {C}^{(1)}_{n-1}$, $\mathrm {A}^{(2)}_{2n-2}$, $\mathrm {D}^{(2)}_{n}$
Yuki Kanakubo, Springer Science and Business Media LLC
Letters in Mathematical Physics, 28 May 2023, [Reviewed]

HALF POTENTIAL ON GEOMETRIC CRYSTALS AND CONNECTEDNESS OF CELLULAR CRYSTALS
YUKI KANAKUBO; TOSHIKI NAKASHIMA, Springer Science and Business Media LLC
Transformation Groups, Mar. 2023, [Reviewed]

An algorithm for Berenstein-Kazhdan decoration functions and trails for minuscule representations
Yuki Kanakubo; Gleb Koshevoy; Toshiki Nakashima, Elsevier BV
Journal of Algebra, Oct. 2022, [Reviewed]

Adapted sequences and polyhedral realizations of crystal bases for highest weight modules
Yuki Kanakubo; Toshiki Nakashima, Elsevier BV
Journal of Algebra, May 2021, [Reviewed]

Adapted sequence for polyhedral realization of crystal bases
Yuki Kanakubo; Toshiki Nakashima, Informa UK Limited
Communications in Algebra, Nov. 2020, [Reviewed]

Geometric crystals and cluster ensembles in Kac-Moody setting　　　　　　　　　　　　　　　
Yuki Kanakubo; Toshiki Nakashima
Journal of Geometry and Physics, Mar. 2020, [Reviewed]

Cluster algebras of finite type via Coxeter elements and Demazure Crystals of type A　　　　　　　　　　　　　　　
Yuki Kanakubo; Toshiki Nakashima
Journal of Algebra, Nov. 2019, [Reviewed]

Cluster algebras of finite type via Coxeter elements and Demazure Crystals of type B,C,D　　　　　　　　　　　　　　　
Yuki Kanakubo
Journal of Geometry and Physics, Mar. 2019, [Reviewed]

Cluster Variables on Double Bruhat Cells G^{u,e} of Classical Groups and Monomial Realizations of Demazure Crystals　　　　　　　　　　　　　　　
Yuki Kanakubo; Toshiki Nakashima
International Mathematics Research Notices, Jun. 2018, [Reviewed]

Explicit Forms of Cluster Variables on Double Bruhat Cells of Type C
Yuki Kanakubo; Toshiki Nakashima, Let G = Sp2r(C) be a simply connected simple algebraic group over C of type Cr, B and B_ its two opposite Borel subgroups, and W the associated Weyl group. For u, v is an element of W, it is known that the coordinate ring C[G(u,v)] of the double Bruhat cell G(u,v) = BuB U B_vB_ is isomorphic to an upper cluster algebra (A) over bar (i)c and the generalized minors Delta(k; i) are the cluster variables of C[G(u,v)][5]. In the case v = e, we shall describe the generalized minor Delta(k; i) explicitly., TOKYO JOURNAL MATHEMATICS EDITORIAL OFFICE ACAD CENTER
TOKYO JOURNAL OF MATHEMATICS, Mar. 2017, [Reviewed]

Explicit forms of cluster variables on double Bruhat cells G^{u,e} of type B
Yuki Kanakubo, Let G be a simply connected simple algebraic group over C of type B-r, B and B- be its two opposite Borel subgroups, and W be the associated Weyl group. For u, vW, it is known that the coordinate ring C[G(u,v)] of the double Bruhat cell G(u,v) = BuB boolean OR B(-)vB(-) is isomorphic to an upper cluster algebra (A) over bar (i)(C) and generalized minors Delta(k; i) are the cluster variables of C[G(u,v)][1]. It is also shown that C[G(u,v)] have a structure of cluster algebra [6]. In the case v = e, we shall describe the generalized minor Delta(k; i) explicitly., TAYLOR & FRANCIS INC
COMMUNICATIONS IN ALGEBRA, Mar. 2017, [Reviewed]

Cluster Variables on Certain Double Bruhat Cells of Type (u, e) and Monomial Realizations of Crystal Bases of Type A
Yuki Kanakubo; Toshi Nakashima, Let C be a simply connected simple algebraic group over C, B and B_ be two opposite Borel subgroups in C and W be the Weyl group. For u, v is an element of W, it is known that the coordinate ring C[C-u,C- v] of the double Bruhat cell C-u,C- v = BuB boolean AND B_vB_ is isomorphic to an upper cluster algebra (A) over bar( i)(C) and the generalized minors {Delta(k; i)} are the cluster variables belonging to a given initial seed in C[C-u,C- v] [Berenstein A., Fomin S., Zelevinsky A., Duke Math. J. 126 (2005), 1-52]. In the case C = SLr+1 (C), v = e and some special u is an element of W, we shall describe the generalized minors {Delta(k; i)} as summations of monomial realizations of certain Demazure crystals., NATL ACAD SCI UKRAINE, INST MATH
SYMMETRY INTEGRABILITY AND GEOMETRY-METHODS AND APPLICATIONS, Apr. 2015, [Reviewed]

Explicit forms of polyhedral realizations of crystal bases associated with adapted sequences　　　　　　　　　　　　　　　
Yuki Kanakubo; Toshiki Nakashima
Toyama Mathematical Journal, 2021, [Reviewed]

Finite type cluster algebras and Demazure crystals (リー型の組合せ論)
Kanakubo Yuki
数理解析研究所講究録, Jul. 2017

Cluster Variables on Double Bruhat Cells and Monomial Realizations of Crystal Bases (New Developments of Representation Theory and Harmonic Analysis)
Kanakubo Yuki
RIMS Kokyuroku, Nov. 2014

Mathematical science　　　　　　　　　　　　　　　
Yuki Kanakubo, Contributor
サイエンス社, Mar. 2025

Inequalities defining polyhedral realizations and monomial realizations of crystal bases
Yuki Kanakubo
Combinatorics and Arithmetic for Physics : special days, 22 Nov. 2024, [Invited]
20241120, 20241122

Polyhedral realizations and Young walls of classical affine types　　　　　　　　　　　　　　　
Yuki Kanakubo
"Crystal Bases and Then..." - Conference in honor of Toshiki Nakashima's 60th birthday -, 12 Jul. 2024, [Invited]
20240709, 20240712

An algorithm for Berenstein-Kazhdan decoration functions on classical groups　　　　　　　　　　　　　　　
Yuki Kanakubo
Cologne Algebra and Representation Theory Seminar, 12 Dec. 2023, [Invited]
20231212, 20231212

Polyhedral realizations for crystal bases and extended Young diagrams of affine type A^{(1)}_{n-1}　　　　　　　　　　　　　　　
Yuki Kanakubo
MPI-Oberseminar, 23 Nov. 2023
20231123, 20231123

Inequalities defining polyhedral realizations of affine types and extended Young diagrams　　　　　　　　　　　　　　　
Yuki Kanakubo
Combinatorics and Arithmetic for Physics: special days, Tenth Anniversary Edition, 17 Nov. 2023, [Invited]
20231115, 20231117

Polyhedral realizations and extended Young diagrams of several classical affine types　　　　　　　　　　　　　　　
Yuki Kanakubo
Sophia University Mathematics Colloquium, 21 Jul. 2023, [Invited]

An algorithm for Berenstein-Kazhdan decoration functions on classical groups　　　　　　　　　　　　　　　
Yuki Kanakubo
Tokyo Tech Representation Theory Seminar, 14 Jul. 2023, [Invited]

Polyhedral realizations and extended Young diagrams of several classical affine types
Yuki Kanakubo
Physical Algebra and Combinatorics Seminar, 24 Mar. 2023, [Invited]

An algorithm for Berenstein-Kazhdan decoration functions on classical groups
Yuki Kanakubo
Algebraic Lie Theory and Representation Theory (ALTReT) 2022, 26 May 2022
20220523, 20220527

An algorithm for Berenstein-Kazhdan decoration functions for minuscule representations
Yuki Kanakubo
MSJ Spring Meeting 2022, 30 Mar. 2022
20220328, 20220331

Polyhedral realizations and extended Young diagrams, Young walls of several classical affine types
Yuki Kanakubo
RIMS workshop, Combinatorial Representation Theory and Connections with Related Fields, 20 Oct. 2021
20211018, 20211022

The inequalities defining polyhedral realizations and monomial realizations of crystal bases　　　　　　　　　　　　　　　
Yuki Kanakubo
組合せ論的表現論の最近の進展, 08 Oct. 2020, [Invited]
20201005, 20201008

Adapted Sequence for Polyhedral Realization of Crystal Bases　　　　　　　　　　　　　　　
Yuki Kanakubo
The 2nd Meeting for Study of Number theory, Hopf algebras and related topics, 16 Feb. 2020, [Invited]
20200215, 20200218

Adapted Sequence for Polyhedral Realization of Crystal Bases　　　　　　　　　　　　　　　
Yuki Kanakubo
Representation Theory and its Combinatorial Aspects, 28 Oct. 2019
20191028, 20191031

Cluster theory on double Bruhat cells and crystal bases　　　　　　　　　　　　　　　
Yuki Kanakubo
Cluster Algebras 2019, 14 Jun. 2019, [Invited]

Adapted Sequence for Polyhedral Realization of Crystal Bases　　　　　　　　　　　　　　　
Yuki Kanakubo
Algebraic Lie Theory and Representation Theory 2019, May 2019

Positivity condition of Polyhedral realizations of crystal bases　　　　　　　　　　　　　　　
Yuki Kanakubo
Crystals and Their Generalizations, 26 Mar. 2019, [Invited]

Positivity condition of Polyhedral realizations of crystal bases　　　　　　　　　　　　　　　
Yuki Kanakubo
Meeting of crystal basis and quantum algebras and superalgebras, 14 Nov. 2018, [Invited]

Cluster algebra structures of the coordinate rings and crystal bases　　　　　　　　　　　　　　　
Yuki Kanakubo
Okayama University of Science, 『第8回 半田山・幾何・代数セミナー』, 11 Jan. 2018, [Invited]

Cluster algebras of finite type and crystal bases　　　　　　　　　　　　　　　
Yuki Kanakubo
Infinite Analysis 17, Algebraic and Combinatorial Aspects in Integrable Systems, 06 Dec. 2017, [Invited]

Cluster algebras of finite type via a Coxeter element and Demazure Crystals　　　　　　　　　　　　　　　
Yuki Kanakubo
Algebraic Analysis and Representation Theory, Jun. 2017

Cluster variables on double Bruhat cells of classical groups and crystal bases　　　　　　　　　　　　　　　
Yuki Kanakubo
筑波大学数学特別セミナー, 04 Nov. 2016, [Invited]

Finite type cluster algebras and Demazure crystals　　　　　　　　　　　　　　　
Yuki Kanakubo
RIMS 研究集会「リー型の組合せ論」, 05 Oct. 2016

Cluster variables on double Bruhat cells of classical groups and crystal bases　　　　　　　　　　　　　　　
Yuki Kanakubo
第2回 Algebraic Lie Theory and Representation Theory, Jun. 2016

Cluster variables on double Bruhat cells of classical groups and crystal bases　　　　　　　　　　　　　　　
Yuki Kanakubo
MSJ Spring Meeting 2016, Mar. 2016

古典群のdouble Bruhat cell 上のクラスター変数と結晶基底　　　　　　　　　　　　　　　
Yuki Kanakubo
第12回 数学総合若手研究集会, Mar. 2016

結晶基底と，Double Bruhat cell 上の座標環のクラスター変数　　　　　　　　　　　　　　　
Yuki Kanakubo
上智大学 数学談話会, Jul. 2015, [Invited]

Cluster Variables on Double Bruhat Cells and Monomial Realizations of Crystal Bases　　　　　　　　　　　　　　　
Yuki Kanakubo
MSJ Autumn Meeting 2014, Sep. 2014

Cluster Variables on Double Bruhat Cells and Monomial Realizations of Crystal Bases　　　　　　　　　　　　　　　
Yuki Kanakubo
RIMS 研究集会「表現論と調和解析の新たな進展」, Jun. 2014

Cluster Variables on Double Bruhat Cells and Monomial Realizations of Crystal Bases　　　　　　　　　　　　　　　
Yuki Kanakubo
第17回代数群と量子群の表現論, Jun. 2014

Exercise in Linear algebra S　　　　　　　　　　　　　　　
May 2023 - Aug. 2023
University of Tsukuba

Math literacy 2　　　　　　　　　　　　　　　
Jun. 2022 - Jul. 2022
University of Tsukuba

Math literacy 1　　　　　　　　　　　　　　　
Apr. 2022 - May 2022
University of Tsukuba

Exercise in linear algebra F　　　　　　　　　　　　　　　
Nov. 2021 - Feb. 2022
University of Tsukuba

Math literacy 2　　　　　　　　　　　　　　　
Jun. 2021 - Jul. 2021
University of Tsukuba

Math literacy 1　　　　　　　　　　　　　　　
Apr. 2021 - May 2021
University of Tsukuba

Exercise in linear algebra F　　　　　　　　　　　　　　　
Nov. 2020 - Feb. 2021
University of Tsukuba

Linear algebra I　　　　　　　　　　　　　　　
Sep. 2020 - Sep. 2020
University of Tsukuba

Calculus I　　　　　　　　　　　　　　　
Sep. 2020 - Sep. 2020
University of Tsukuba

日本数学会