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Toshiki Nakashima

Publications and source records attributed to Toshiki Nakashima.

At least 19 recordsLinked to original sources
arXiv2025

Characterization of the unit object in localized quantum unipotent category

For the quiver Hecke algebra $R$, let $R\hbox{-gmod}$ be the category of finite-dimensional graded $R$-modules, and let $\widetilde{R\hbox{-gmod}[w]}$ be the localization of $R\hbox{-gmod}$. Kashiwara and the second author showed the set of equivalence classes of simple objects up to grading shifts $\mathrm{Irr}(\widetilde{R\hbox{-gmod}[w]})$ in $\widetilde{R\hbox{-gmod}[w]}$ has a crystal structure, and $\mathrm{Irr}(\widetilde{R\hbox{-gmod}[w]})$ is isomorphic to the so-called cellular crystal $\mathbb B_{\mathbf i}$. This isomorphism induces a function $\varepsilon_i^*$ on $\mathbb B_{\mathbf i}$. We give an explicit formula of $\varepsilon_i^*$, and using this formula, we give a characterization of the unit object of $\widetilde{R\hbox{-gmod}[w]}$ for the case of classical finite types.

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arXiv2025

Crystal Structure of Localized Quantum Unipotent Coordinate Category

A localized quantum unipotent coordinate category $\widetilde{\mathscr{C}_w}$ associated with a Weyl group element $w$ is a rigid monoidal category which is obtained by applying the localization process to a subcategory of the category of finite-dimensional graded modules of a quiver Hecke algebra. We shall show that the family of the isomorphism classes (up to grading shifts) of simple objects in $\widetilde{\mathscr{C}_w}$ possesses a crystal structure and it is isomorphic to the cellular crystal associated with $w$. As an application of this result, we shall show the connectedness of the crystal graph of an arbitrary cellular crystal.

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arXiv2025

Decomposition Theorem for Product of Fundamental Crystals in Monomial Realization of type $C_n$

We consider a product of fundamental crystals of type $C_n$ in monomial realization, where the product means a natural product of Laurent monomials, not a tensor product. Then we shall show that the product still holds a crystal structure and describe how it is decomposed into irreducible crystals, which is, in general, different from the decomposition rule for the tensor product of the fundamental crystals.

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arXiv2025

Products of Kirillov-Reshetikhin modules and maximal green sequences

We show that a $q$-character of a Kirillov-Reshetikhin module (KR modules) for untwisted quantum affine algebras of simply laced types $A_n^{(1)}$, $D_n^{(1)}$, $E_6^{(1)}$, $E_7^{(1)}$, $E_8^{(1)}$ might be obtained from a specific cluster variable of a seed obtained by applying a maximal green sequence to the initial (infinite) quiver of the Hernandez-Leclerc cluster algebra. For a collection of KR-modules with nested supports, we show an explicit construction of a cluster seed, which has cluster variables corresponding to the $q$-characters of KR-modules of such a collection. We prove that the product of KR-modules of such a collection is a simple module. We also construct cluster seeds with cluster variables corresponding to $q$-characters of KR-modules of some non-nested collections. We make a conjecture that tensor products of KR-modules for such non-nested collections are simple. We show that the cluster Donaldson-Thomas transformations for double Bruhat cells for $ADE$ types can be computed using $q$-characters of KR-modules.

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arXiv2022

Categorified Crystal Structure on Localized Quantum Coordinate Rings

For the quiver Hecke algebra $R$ associated with a simple Lie algebra, let $R$-gmod be the category of finite-dimensional graded $R$-modules. It is well-known that it categorifies the unipotent quantum coordinate ring. The localization of $R$-gmod has been defined in [12]. Its Grothendieck ring defines the localized (unipotent) quantum coordinate ring. We shall give a certain crystal structure on the localized quantum coordinate ring by regarding the set of self-dual simple objects in localized $R$-gmod. We also give the isomorphism of crystals to the cellular crystal for an arbitrary reduced word of the longest Weyl group element. This result can be seen as a localized version of the categorification for the crystal of the nilpotent half of quantum algebra by Lauda and Vazirani.

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arXiv2022

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

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 $θ^-_{\mathbf{i}}:(\mathbb{C}^{\times})^{l(w_0)}\rightarrow B_{w_0}^-$ for each reduced word $\mathbf{i}$ of the longest element $w_0$ of Weyl group. A rational function $Φ^h_{BK}=\sum_{i\in I}Δ_{w_0Λ_i,s_iΛ_i}$ on $B_{w_0}^-$ is called a half-potential, where $Δ_{w_0Λ_i,s_iΛ_i}$ is a generalized minor. Computing $Φ^h_{BK}\circ θ^-_{\mathbf{i}}$ explicitly, we get an explicit form of string cone or polyhedral realization of $B(\infty)$ for the finite dimensional simple Lie algebra $\mathfrak{g}={\rm Lie}(G)$. In this paper, for an arbitrary reduced word $\mathbf{i}$, we give an algorithm to compute the summand $Δ_{w_0Λ_i,s_iΛ_i}\circ θ^-_{\mathbf{i}}$ of $Φ^h_{BK}\circ θ^-_{\mathbf{i}}$ in the case $i\in I$ satisfies that for any weight $μ$ of $V(-w_0Λ_i)$ and $t\in I$, it holds $\langle h_t,μ\rangle\in\{2,1,0,-1,-2\}$. In particular, if $\mathfrak{g}$ is of type ${\rm A}_n$, ${\rm B}_n$, ${\rm C}_n$ or ${\rm D}_n$ then all $i\in I$ satisfy this condition so that one can completely calculate $Φ^h_{BK}\circ θ^-_{\mathbf{i}}$. We will also prove that our algorithm works in the case $\mathfrak{g}$ is of type ${\rm G}_2$.

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arXiv2022

Half Potential on Geometric Crystals and Connectedness of Cellular Crystals

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. In particular, we show that the tropicalization of the half-decorated geometric crystal on the big Bruhat cell(=$B^-_{w_0}:=B^-\cap U\bar w_0 U$) is isomorphic to the crystal $B(\infty)$ of the nilpotent subalgebra of quantum group $U_q^-(\mathfrak g)$. As an application, we shall show that any cellular crystal associated with a reduced word is connected in the sense of a crystal graph.

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arXiv2021

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

The polyhedral realizations for crystal bases of the integrable highest weight modules of $U_q(\mathfrak{g})$ have been introduced in ([T.Nakashima, J. Algebra, vol.219, no. 2, (1999)]), which describe the crystal bases as sets of lattice points in the infinite $\mathbb{Z}$-lattice $\mathbb{Z}^{\infty}$ given by some system of linear inequalities, where $\mathfrak{g}$ is a symmetrizable Kac-Moody Lie algebra. To construct the polyhedral realization, we need to fix an infinite sequence $ι$ from the indices of the simple roots. If the pair ($ι$,$λ$) ($λ$: a dominant integral weight) satisfies the `ample' condition then there are some procedure to calculate the sets of linear inequalities. In this article, we show that if $ι$ is an adapted sequence (defined in our paper [Y.Kanakubo, T.Nakashima, arXiv:1904.10919]) then the pair ($ι$, $λ$) satisfies the ample condition for any dominant integral weight $λ$ in the case $\mathfrak{g}$ is a classical Lie algebra. Furthermore, we reveal the explicit forms of the polyhedral realizations of the crystal bases $B(λ)$ associated with arbitrary adapted sequences $ι$ in terms of column tableaux. As an application, we will give a combinatorial description of the function $\varepsilon_i^*$ on the crystal base $B(\infty)$.

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arXiv2021

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

For a simply connected connected simple algebraic group $G$, a cell $B_{w_0}^-=B^-\cap U\overline{w_0}U$ is a geometric crystal with a positive structure $θ_{\textbf{i}}^-:(\mathbb{C}^{\times})^{l(w_0)}\rightarrow B_{w_0}^-$. Applying the tropicalization functor to a rational function $Φ^h_{BK}=\sum_{i\in I}Δ_{w_0Λ_i,s_iΛ_i}$ called the half decoration on $B_{w_0}^-$, one can realize the crystal $B(\infty)$ in $\mathbb{Z}^{l(w_0)}$. By computing $Φ^h_{BK}$, we get an explicit form of $B(\infty)$ in $\mathbb{Z}^{l(w_0)}$. In this paper, we give an algorithm to compute $Δ_{w_0Λ_i,s_iΛ_i}\circ θ_{\textbf{i}}^-$ explicitly for $i\in I$ such that $V(Λ_i)$ is a minuscule representation of $\mathfrak{g}={\rm Lie}(G)$. In particular, the algorithm works for all $i\in I$ if $\mathfrak{g}$ is of type ${\rm A}_n$. The algorithm computes a directed graph $DG$, called a decoration graph, whose vertices are labelled by all monomials in $Δ_{w_0Λ_i,s_iΛ_i}\circ θ_{\textbf{i}}^-(t_1,\cdots,t_{l(w_0)})$. The decoration graph has some properties similar to crystal graphs of minuscule representations. We also verify that the algorithm works in some other cases, for example, the case $\mathfrak{g}$ is of type ${\rm G}_2$ though $V(Λ_i)$ is non-minuscule.

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arXiv2021

Adapted Sequence for Polyhedral Realization of Crystal Bases

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 crystal base $B(\infty)$ as a polyhedral convex cone in the infinite $\mathbb{Z}$-lattice $\mathbb{Z}^{\infty}$. To construct the polyhedral realization, we need to fix an infinite sequence $ι$ from the indices of the simple roots. According to this $ι$, one has certain set of linear functions defining a polyhedral convex cone and under the `positivity condition' on $ι$, it has been shown that the polyhedral convex cone is isomorphic to the crystal base $B(\infty)$. To confirm the positivity condition for a given $ι$, we need to obtain the whole feature of the set of linear functions, which requires, in general, a bunch of explicit calculations. In this article, we introduce the notion of the adapted sequence and show that if $ι$ is an adapted sequence then the positivity condition holds for classical Lie algebras. Furthermore, we reveal the explicit forms of the polyhedral realizations associated with arbitrary adapted sequences $ι$ in terms of column tableaux.

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arXiv2018

Geometric crystals and Cluster ensembles in Kac-Moody setting

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 birational maps between `cluster tori' $\mathcal{X}_Σ$ (resp. $\mathcal{A}_Σ$) and $G_{\rm Ad}^{u,e}$ (resp. $G^{u,e}$), and they are extended to regular maps from cluster $\mathcal{X}$ (resp. $\mathcal{A}$) -varieties to $G_{\rm Ad}^{u,e}$ (resp. $G^{u,e}$). The aim of this article is to construct certain positive geometric crystal structures on the cluster tori $\mathcal{X}_Σ$ and $\mathcal{A}_Σ$ by presenting their explicit formulae. In particular, the geometric crystal structures on the tori $\mathcal{A}_Σ$ are obtained by applying the twist map. As a corollary, we see the sets of $\mathbb{Z}^T$-valued points of the cluster varieties have plural structures of crystals.

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arXiv2017

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

Let $G=Sp_{2r}({\mathbb C})$ be a simply connected simple algebraic group over $\mathbb{C}$ of type $C_r$, $B$ and $B_-$ be its two opposite Borel subgroups, and $W$ be the associated Weyl group. For $u$, $v\in W$, it is known that the coordinate ring ${\mathbb C}[G^{u,v}]$ of the double Bruhat cell $G^{u,v}=BuB\cap B_-vB_-$ is isomorphic to an upper cluster algebra $\overline{\mathcal{A}}(\textbf{i})_{\mathbb C}$ and the generalized minors $Δ(k;\textbf{i})$ are the cluster variables of ${\mathbb C}[G^{u,v}]$[Berenstein A., Fomin S., Zelevinsky A., Duke Math. J. 126 (2005), 1-52, math.RT/0305434]. In the case $v=e$, we shall describe the generalized minor $Δ(k;\textbf{i})$ explicitly.

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arXiv2017

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

Let $G$ be a simply connected simple algebraic group over $\mathbb{C}$, $B$ and $B_-$ be its two opposite Borel subgroups. For two elements $u$, $v$ of the Weyl group $W$, it is known that the coordinate ring ${\mathbb C}[G^{u,v}]$ of the double Bruhat cell $G^{u,v}=BuB\cap B_-vB_-$ is isomorphic to a cluster algebra $\mathcal{A}(\textbf{i})_{\mathbb C}$ [arXiv:math/0305434, arXiv:1602.00498]. In the case $u=e$, $v=c^2$ ($c$ is a Coxeter element), the algebra ${\mathbb C}[G^{e,c^2}]$ has only finitely many cluster variables. In this article, for $G={\rm SL}_{r+1}(\mathbb{C})$, we obtain explicit forms of all the cluster variables in $\mathbb{C}[G^{e,c^2}]$ by considering its additive categorification via preprojective algebras, and describe them in terms of monomial realizations of Demazure crystals.

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arXiv2017

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

Let $G$ be a simply connected simple algebraic group over $\mathbb{C}$, $B$ and $B_-$ its two opposite Borel subgroups, and $W$ the associated Weyl group. It is shown that the coordinate ring ${\mathbb C}[G^{u,v}]$ ($u$, $v\in W$) of the double Bruhat cell $G^{u,v}=BuB\cap B_-vB_-$ is isomorphic to the cluster algebra ${\mathcal{A}}(\textbf{i})_{\mathbb C}$ and the initial cluster variables of ${\mathbb C}[G^{u,v}]$ are the generalized minors $Δ(k;\textbf{i})$ by Berenstein, Fomin, Zelevinsky, Goodearl and Yakimov. In the case that a classical group $G$ is of type ${\rm B}_r$, ${\rm C}_r$ or ${\rm D}_r$, we shall describe the non-trivial last $r$ initial cluster variables $\{Δ(k;\textbf{i})\}_{(m-2)r<k\leq (m-1)r}$ ($m$ is some positive integer) of the cluster algebra $\mathbb{C}[L^{u,e}]$ in terms of monomial realization of Demazure crystals, where $L^{u,e}$ is the reduced double Bruhat cell of type $(u,e)$. The relation between $Δ(k;\textbf{i})$ on $G^{u,e}$ and on $L^{u,e}$ is described as well. We also present the corresponding results for type ${\rm A}_r$ though the results for all initial cluster variables have been obtained by ourselves.

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arXiv2016

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

Let $\mathfrak g$ be an affine Lie algebra with index set $I = \{0, 1, 2, \cdots , n\}$ and ${\mathfrak g}^L$ be its Langlands dual. It is conjectured by Kashiwara et al.([16]) that for each $k \in I \setminus \{0\}$ the affine Lie algebra $\mathfrak g$ has a positive geometric crystal whose ultra-discretization is isomorphic to the limit of certain coherent family of perfect crystals for ${\mathfrak g}^L$. Motivated by this conjecture we construct a positive geometric crystal for the affine Lie algebra ${\mathfrak g}= A^{(1)}_n$ for each Dynkin index $k\in I\setminus\{0\}$ and show that its ultra-discretization is isomorphic to the limit of a coherent family of perfect crystals for $A^{(1)}_n$ given by Okado et al.([29]). In the process we develop and use some lattice-path combinatorics.

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arXiv2015

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

Let $G$ be a simply connected simple algebraic group over $\mathbb{C}$, $B$ and $B_-$ be two opposite Borel subgroups in $G$ and $W$ be the Weyl group. For $u$, $v\in W$, it is known that the coordinate ring ${\mathbb C}[G^{u,v}]$ of the double Bruhat cell $G^{u,v}=BuB\cap B_-vB_-$ is isomorphic to an upper cluster algebra $\bar{\mathcal A}({\bf i})_{\mathbb C}$ and the generalized minors $\{Δ(k;{\bf i})\}$ are the cluster variables belonging to a given initial seed in ${\mathbb C}[G^{u,v}]$ [Berenstein A., Fomin S., Zelevinsky A., Duke Math. J. 126 (2005), 1-52, math.RT/0305434]. In the case $G={\rm SL}_{r+1}({\mathbb C})$, $v=e$ and some special $u\in W$, we shall describe the generalized minors $\{Δ(k;{\bf i})\}$ as summations of monomial realizations of certain Demazure crystals.

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