Toshiki Nakashima

activity

19922025

most citedQuantum affine symmetry in vertex models

78 citations · 125 across the 47 of their papers we have counts for

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40 papers · 1 filter

math.QA2022

An algorithm for Berenstein-Kazhdan decoration functions and trails for classical Lie algebras

Yuki Kanakubo, Gleb Koshevoy, Toshiki Nakashima

For a simply connected connected simple algebraic group $G$ , it is known that a variety $B_{w_{0}}^{-} := B^{-} \cap U w_{0} U$ has a geometric crystal structure with a positive stru…

math.QA2021

An algorithm for Berenstein-Kazhdan decoration functions and trails for minuscule representations

Yuki Kanakubo, Gleb Koshevoy, Toshiki Nakashima

For a simply connected connected simple algebraic group $G$ , a cell $B_{w_{0}}^{-} = B^{-} \cap U w_{0} U$ is a geometric crystal with a positive structure $θ_{\textbf{i}}^-:(\mathbb…

math.QA2020

Adapted Sequences and Polyhedral Realizations of Crystal Bases for highest weight modules

Yuki Kanakubo, Toshiki Nakashima

The polyhedral realizations for crystal bases of the integrable highest weight modules of $U_{q} (g)$ have been introduced in ([T.Nakashima, J. Algebra, vol.219, no. 2, (19…

math.QA2019★ 3 cited

Half Potential on Geometric Crystals and Connectedness of Cellular Crystals

Yuki Kanakubo, Toshiki Nakashima

For any simple complex algebraic group, we define upper/lower half-decorated geometric crystals and show that their tropicalization will be upper/lower normal Kashiwara's crystals.…

math.QA2019

Adapted Sequence for Polyhedral Realization of Crystal Bases

Yuki Kanakubo, Toshiki Nakashima

The polyhedral realization of crystal base has been introduced by A.Zelevinsky and the second author([T.Nakashima, A.Zelevinsky, Adv. Math. 131, no. 1 (1997)]), which describe the…

math.QA2018

Decomposition Theorem for Product of Fundamental Crystals in Monomial Realization

Manal Alshuqayr, Toshiki Nakashima

We consider a product of fundamental crystals in monomial realization of type A. Then we shall show that the product holds crystal structure and describe how it is decomposed into…

math.QA2018★ 1 cited

Geometric crystals and Cluster ensembles in Kac-Moody setting

Yuki Kanakubo, Toshiki Nakashima

For a Kac-Moody group $G$ , double Bruhat cells $G^{u, e}$ ( $u$ is a Weyl group element) have positive geometric crystal structures. In arXiv:1210.2533, it is shown that there exist…

math.QA2017

Cluster algebras of finite type via a Coxeter element and Demazure Crystals of type A

Yuki Kanakubo, Toshiki Nakashima

Let $G$ be a simply connected simple algebraic group over $C$ , $B$ and $B_{-}$ be its two opposite Borel subgroups. For two elements $u$ , $v$ of the Weyl group $W$ , it is kn…

math.QA2017

Cluster algebras of finite type via a Coxeter element and Demazure Crystals of type B,C,D

Yuki Kanakubo

For a classical group $G$ and a Coxeter element $c$ of the Weyl group, it is known that the coordinate ring $C [G^{e, c^{2}}]$ of the double Bruhat cell $G^{e,c^2}:=B\cap B_-c…

math.QA2016★ 2 cited

Affine Geometric Crystal of $A_{n}^{(1)}$ and Limit of Kirillov-Reshetikhin Perfect Crystals

Kailash C. Misra, Toshiki Nakashima

Let $g$ be an affine Lie algebra with index set $I = {0, 1, 2, \dots, n}$ and $g^{L}$ be its Langlands dual. It is conjectured by Kashiwara et al.([16]) tha…

math.QA2016★ 2 cited

Cluster Variables on Double Bruhat Cells $G^{u, e}$ of Classical Groups and Monomial Realizations of Demazure Crystals

Yuki Kanakubo, Toshiki Nakashima

Let $G$ be a simply connected simple algebraic group over $C$ , $B$ and $B_{-}$ its two opposite Borel subgroups, and $W$ the associated Weyl group. It is shown that the coor…

math.QA2016★ 1 cited

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 $…

math.QA2015★ 3 cited

Explicit Forms of Cluster Variables on Double Bruhat Cells G^{u,e} of type C

Yuki Kanakubo, Toshiki Nakashima

Let $G = S p_{2 r} (C)$ be a simply connected simple algebraic group over $C$ of type $C_{r}$ , $B$ and $B_{-}$ be its two opposite Borel subgroups, and $W$ be the associa…

math.QA2014★ 9 cited

Cluster Variables on Certain Double Bruhat Cells of Type $(u, e)$ and Monomial Realizations of Crystal Bases of Type A

Yuki Kanakubo, Toshiki Nakashima

Let $G$ be a simply connected simple algebraic group over $C$ , $B$ and $B_{-}$ be two opposite Borel subgroups in $G$ and $W$ be the Weyl group. For $u$ , $v \in W$ , it is kno…

math.QA2013

Decorated Geometric Crystals and Polyhedral Realizations of type D_n

Toshiki Nakashima

This is a continuation of [15, 16]. We shall show that for type D_n the realization of crystal bases obtained from the decorated geometric crystals in [2] coincides with the polyhe…

math.QA2013

Decorations on Geometric Crystals and Monomial Realizations of Crystal Bases for Classical Groups

Toshiki Nakashima

We shall describe explicitly the decoration functions for certain decorated geometric crystals of classical groups and we shall show that they are represented in terms of monomial…

math.QA2012

$A_{n}^{(1)}$ -Geometric Crystal corresponding to Dynkin index $i = 2$ and its ultra-discretization

Kailash C. Misra, Toshiki Nakashima

Let $g$ be an affine Lie algebra with index set $I = {0, 1, 2, ..., n}$ and $g^{L}$ be its Langlands dual. It is conjectured that for each $i \in I ∖ {0}$ the affine Lie a…

math.QA2012

Decorated Geometric Crystals, Polyhedral and Monomial Realizations of Crystal Bases

Toshiki Nakashima

We shall show that for type $A_{n}$ the realization of crystal bases obtained from the decorated geometric crystals intorduced by Berenstein and Kazhdan coincides with our polyhedral…

math.QA2010

RSK type correspondence of Pictures and Littlewood-Richardson Crystals

Toshiki Nakashima, Miki Shimojo

We present a Robinson-Schensted-Knuth type one-to-one correspondence between the set of pictures and the set of pairs of Littlewood-Richardson crystals.

math.QA2010

Ultra-discretization of the D_4^3-Geometric Crystals to the G_2^1-Perfect Crystals

Mana Igarashi, Kailash C. Misra, Toshiki Nakashima

Let g be an affine Lie algebra and g^L be its Langlands dual. It is conjectured that g has a positive geometric crystal whose ultra-discretization is isomorphic to the limit of cer…

math.QA2010

Geometric Crystals on Flag Varieties and Unipotent Subgroups of Classical Groups

Mana Igarashi, Toshiki Nakashima

For a classical simple algebraic group $G$ we obtain the affirmative answer for the conjecture in [8] that there exists an isomorphism between the geometric crystal on the flag var…

math.QA2009

Affine Geometric Crystal of type $D_{4}^{(3)}$

Mana Igarashi, Toshiki Nakashima

We shall realize certain affine geometric crystal of type $D_{4}^{(3)}$ associated with the fundamental representation $W (π_{1})$ explicitly . By its explicit form, we see that it has…

math.QA2009★ 1 cited

Epsilon Systems on Geometric Crystals of Type $A_{n}$

Toshiki Nakashima

We introduce an epsilon system on a geometric crystal of type $A_{n}$ , which is a certain set of rational functions with some nice properties. We shall show that it is equipped with…

math.QA2009★ 1 cited

Admissible Pictures and Littlewood-Richardson Crystals

Toshiki Nakashima, Miki Shimojo

We present a one-to-one correspondence between the set of admissible pictures and the Littlewood-Richardson crystals. As a simple consequence, we shall show that the set of picture…

An algorithm for Berenstein-Kazhdan decoration functions and trails for classical Lie algebras

An algorithm for Berenstein-Kazhdan decoration functions and trails for minuscule representations

Adapted Sequences and Polyhedral Realizations of Crystal Bases for highest weight modules

Half Potential on Geometric Crystals and Connectedness of Cellular Crystals

Adapted Sequence for Polyhedral Realization of Crystal Bases

Decomposition Theorem for Product of Fundamental Crystals in Monomial Realization

Geometric crystals and Cluster ensembles in Kac-Moody setting

Cluster algebras of finite type via a Coxeter element and Demazure Crystals of type A

Cluster algebras of finite type via a Coxeter element and Demazure Crystals of type B,C,D

Affine Geometric Crystal of An(1)​ and Limit of Kirillov-Reshetikhin Perfect Crystals

Cluster Variables on Double Bruhat Cells Gu,e of Classical Groups and Monomial Realizations of Demazure Crystals

Explicit Forms of Cluster Variables on Double Bruhat Cells G^{u,e} of type B

Explicit Forms of Cluster Variables on Double Bruhat Cells G^{u,e} of type C

Cluster Variables on Certain Double Bruhat Cells of Type (u,e) and Monomial Realizations of Crystal Bases of Type A

Decorated Geometric Crystals and Polyhedral Realizations of type D_n

Decorations on Geometric Crystals and Monomial Realizations of Crystal Bases for Classical Groups

An(1)​-Geometric Crystal corresponding to Dynkin index i=2 and its ultra-discretization

Decorated Geometric Crystals, Polyhedral and Monomial Realizations of Crystal Bases

RSK type correspondence of Pictures and Littlewood-Richardson Crystals

Ultra-discretization of the D_4^3-Geometric Crystals to the G_2^1-Perfect Crystals

Geometric Crystals on Flag Varieties and Unipotent Subgroups of Classical Groups

Affine Geometric Crystal of type D4(3)​

Epsilon Systems on Geometric Crystals of Type An​

Admissible Pictures and Littlewood-Richardson Crystals

Affine Geometric Crystal of $A_{n}^{(1)}$ and Limit of Kirillov-Reshetikhin Perfect Crystals

Cluster Variables on Double Bruhat Cells $G^{u, e}$ of Classical Groups and Monomial Realizations of Demazure Crystals

Cluster Variables on Certain Double Bruhat Cells of Type $(u, e)$ and Monomial Realizations of Crystal Bases of Type A

$A_{n}^{(1)}$ -Geometric Crystal corresponding to Dynkin index $i = 2$ and its ultra-discretization

Affine Geometric Crystal of type $D_{4}^{(3)}$

Epsilon Systems on Geometric Crystals of Type $A_{n}$