Realizations of Crystals

Polyhedral realization of Crystal bases for Generalized Kac-Moody Algebras

In this paper, we give polyhedral realization of the crystal $B(\infty)$ of $U_q^-(\mathfrak g)$ for the generalized Kac-Moody algebras. As applications, we give explicit descriptions of crystals for the generalized Kac-Moody algebras of rank 2, 3 and {\it Monster Lie algebras}.

Polyhedral Realizations of Crystal Bases for Quantum Algebras of Finite Types

Polyhedral realization of crystal bases is one of the methods for describing the crystal base $B(\infty)$ explicitly. This method can be applied to symmetrizable Kac-Moody types. We can also apply this method to the crystal bases $B(λ)$ of integrable highest weight modules and of modified quantum algebras. But, the explicit forms of the polyhedral realizations of crystal bases $B(\infty)$ and $B(λ)$ are only given in the case of arbitrary rank 2, of $A_n$ and of $A^{(1)}_n$. So, we will give the polyhedral realizations of crystal bases $B(\infty)$ and $B(λ)$ for all simple Lie algebras in this…

Young Wall Realization of Crystal Bases for Classical Lie Algebras

In this paper, we give a new realization of crystal bases for finite dimensional irreducible modules over classical Lie algebras. The basis vectors are parameterized by certain Young walls lying between highest weight and lowest weight vectors.

Polyhedral Realizations of Crystal Bases for Modified Quantum Algebras of Type A

We describe the crystal bases of modified quantum algebras and its connected component containing ``zero vector''by the polyhedral realization method for the types A_n and A^(1)_1. We also present the explicit form of the unique highest weight vector in the connected component.

Decorated Geometric Crystals and Polyhedral Realizations of type D_n

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 polyhedral realizations of crystal bases.

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 …

Polyhedral realization of the crystal bases for extremal weight modules over quantized hyperbolic Kac-Moody algebras of rank 2

Let $\mathfrak{g}$ be a hyperbolic Kac-Moody algebra of rank $2$. We give a polyhedral realization of the crystal basis for the extremal weight module of extremal weight $λ$, where $λ$ is an integral weight whose Weyl group orbit has neither a dominant integral weight nor an antidominant integral weight.

Polyhedral Realizations of Crystal Bases for Modified Quantum Algebras of Arbitrary Rank 2 Cases

We describe the crystal bases of the modified quantum algebras and give the explicit form of the highest (or lowest) weight vector of its connected component $B_0(λ)$ containing the unit element for arbitrary rank 2 cases. We also present the explicit form of $B_0(λ)$ containing the highest (or lowest) weight vector by the polyhedral realization method.

Level 1 Perfect Crystals and Path Realizations of Basic Representations at q=0

We present a uniform construction of level 1 perfect crystals $\mathcal B$ for all affine Lie algebras. We also introduce the notion of a crystal algebra and give an explicit description of its multiplication. This allows us to determine the energy function on $\mathcal B \otimes \mathcal B$ completely and thereby give a path realization of the basic representations at $q=0$ in the homogeneous picture.

Polyhedral Realizations of Crystal Bases and Braid-type Isomorphisms

We review the polyhedral realizations of crystal bases in the former half and in the latter half, we introduce braid-type isomorphisms for some rank 2 finite type crystals. Using this isomorphisms, for semi-simple Lie algebra we can show that polyhedral realizations can be obtained in the lattice of finite rank which coincides with the length of the longest element in the associated Weyl group.

Crystal bases and monomials for $U_q(G_2)$-modules

In this paper, we give a new realization of crystal bases for irreducible highest weight modules over $U_q(G_2)$ in terms of monomials. We also discuss the natural connection between the monomial realization and tableau realization.

Monomial Crystals and Partition Crystals

Recently Fayers introduced a large family of combinatorial realizations of the fundamental crystal for affine sl(n), where the vertices are indexed by certain partitions. He showed that special cases of this construction agree with the Misra-Miwa realization and with Berg's ladder crystal. Here we show that another special case is naturally isomorphic to a realization using Nakajima's monomial crystal.

Polyhedral Realizations of Crystal Bases for Integrable Highest Weight Modules

We give a general way of representing the crystal (base) corresponding to the intgrable highest weight modules of quantum Kac-Moody algebras, which is called polyhedral realizations. This is applied to describe explicitly the crystal bases of integrable highest weight modules for arbitrary rank 2 Kac-Moody algebra cases, the classical A_n-case and the affine A^{(1)}_{n-1}-case.

Lectures on geometric realizations of crystals

These are notes for a lecture series given at the Fields Institute Summer School in Geometric Representation Theory and Extended Affine Lie Algebras, held at the University of Ottawa in June 2009. We give an introduction to the geometric realization of crystal graphs via the quiver varieties of Lusztig and Nakajima. The emphasis is on motivating the constructions through concrete examples. The relation between the geometric construction of crystals and combinatorial realizations using Young tableaux is also discussed.

Geometric and combinatorial realizations of crystal graphs

For irreducible integrable highest weight modules of the finite and affine Lie algebras of type A and D, we define an isomorphism between the geometric realization of the crystal graphs in terms of irreducible components of Nakajima quiver varieties and the combinatorial realizations in terms of Young tableaux and Young walls. For affine type A, we extend the Young wall construction to arbitrary level, describing a combinatorial realization of the crystals in terms of new objects which we call Young pyramids.

Polyhedral Realizations of Crystal Bases for Quantized Kac-Moody Algebras

Let B(\infty) be the crystal corresponding to the nilpotent part of a quantized Kac-Moody algebra. We suggest a general way to represent B(\infty) as the set of integer solutions of a system of linear inequalities. As an application, we treat in a unified manner all Kac-Moody algebras of rank 2 (sharpening the result by Kashiwara), as well as the algebras of types A_n and A_{n-1}^{(1)}.

On the Combinatorics of Crystal Graphs, II. The Crystal Commutator

We present an explicit combinatorial realization of the commutor in the category of crystals which was first studied by Henriques and Kamnitzer. Our realization is based on certain local moves defined by van Leeuwen.

Geometric and Combinatorial Realizations of Crystals of Enveloping Algebras

Kashiwara and Saito have defined a crystal structure on the set of irreducible components of Lusztig's quiver varieties. This gives a geometric realization of the crystal graph of the lower half of the quantum group associated to a simply-laced Kac-Moody algebra. Using an enumeration of the irreducible components of Lusztig's quiver varieties in finite and affine type A by combinatorial data, we compute the geometrically defined crystal structure in terms of this combinatorics. We conclude by comparing the combinatorial realization of the crystal graph thus obtained with other combinatorial mo…

Realization of affine type A Kirillov-Reshetikhin crystals via polytopes

On the polytope defined in Feigin, Fourier, and Littelmann (2011), associated to any rectangle highest weight, we define a structure of an type $A_n$-crystal. We show, by using the Stembridge axioms, that this crystal is isomorphic to the one obtained from Kashiwara's crystal bases theory. Further we define on this polytope a bijective map and show that this map satisfies the properties of a weak promotion operator. This implies in particular that we provide an explicit realization of Kirillov-Reshetikhin crystals for the affine type $A^{(1)}_n$ via polytopes.

Combinatorial realizations of crystals via torus actions on quiver varieties

Consider Kashiwara's crystal associated to a highest weight representation of a symmetric Kac-Moody algebra. There is a geometric realization of this object using Nakajima's quiver varieties, but in many particular cases it can also be realized by elementary combinatorial methods. Here we propose a framework for extracting combinatorial realizations from the geometric picture: We construct certain torus actions on the quiver varieties and use Morse theory to index the irreducible components by connected components of the subvariety of torus fixed points. We then discuss the case of affine sl(n…

Newton-Okounkov convex bodies of Schubert varieties and polyhedral realizations of crystal bases

A Newton-Okounkov convex body is a convex body constructed from a projective variety with a valuation on its homogeneous coordinate ring; this is deeply connected with representation theory. For instance, the Littelmann string polytopes and the Feigin-Fourier-Littelmann-Vinberg polytopes are examples of Newton-Okounkov convex bodies. In this paper, we prove that the Newton-Okounkov convex body of a Schubert variety with respect to a specific valuation is identical to the Nakashima-Zelevinsky polyhedral realization of a Demazure crystal. As an application of this result, we show that Kashiwara'…

关于晶体基的单项式实现与多面体实现的一个猜想

A conjecture on monomial realizations and polyhedral realizations for crystal bases

本文提出并部分证明了一个猜想，该猜想建立了可对称化 Kac-Moody 代数晶体基的多面体锥显式形式与单项式实现之间的联系，并在经典李代数、秩2 Kac-Moody 代数及经典仿射李代数情形下验证了其正确性。

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 $…

Nakajima monomials and crystals for special linear Lie algebras

We present explicit descriptions of the crystals $\mathcal{B}(\infty)$ and $\mathcal{B}(λ)$ over special linear Lie algebras in the language of \emph{extended Nakajima monomials}. There is a natural correspondence between the monomial description and Young tableau realization, which is another realization of crystals $\mathcal{B}(\infty)$ and $\mathcal{B}(λ)$.

源于平移的 E 型 Kirillov-Reshetikhin 晶体的组合实现

A combinatorial realization of Kirillov-Reshetikhin crystals for type E arising from translations

本文通过量子幂零子代数晶体扩展为仿射晶体，给出了极小节点下 E_n^(1) 型 Kirillov-Reshetikhin 晶体 B^{r,s} 的组合实现。

E型量子仿射代数1级不可约最高权与Fock空间晶体的Young wall实现

Young wall realizations of level 1 irreducible highest weight and Fock space crystals of quantum affine algebras in type E

本文为$E_{6}^{(1)}$、$E_{7}^{(1)}$和$E_{8}^{(1)}$型量子仿射代数的1级不可约最高权与Fock空间表示晶体基构造Young wall模型，通过彩色块的堆叠与增减描述晶体结构。

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 c…

Nakajima箭图簇、仿射晶体与Auslander-Reiten箭图的组合学

Nakajima quiver varieties, affine crystals and combinatorics of Auslander-Reiten quivers

本文建立了A型和D型单李代数晶体基两种实现间的显式晶体同构，给出了Lusztig箭图簇不可约分支的同调描述，并通过A型提升算子实现了Kirillov-Reshetikhin晶体的几何构造。

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 th…

Homological description of crystal structures on Lusztig's quiver varieties

Using methods of homological algebra, we obtain an explicit crystal isomorphism between two realizations of crystal bases of the lower part of the quantized enveloping algebra of (almost all) finite dimensional simply-laced Lie algebras. The first realization we consider is a geometric construction in terms of irreducible components of certain quiver varieties established by Kashiwara and Saito. The second is a realization in terms of isomorphism classes of quiver representations obtained by Reineke using Ringel's Hall algebra approach to quantum groups. We show that these two constructions ar…

可积最高权模晶体基的多面体实现以及${\rm A}^{(1)}_{n-1}$、${\rm C}^{(1)}_{n-1}$、${\rm A}^{(2)}_{2n-2}$、${\rm D}^{(2)}_{n}$型组合对象

Polyhedral realizations for crystal bases of integrable highest weight modules and combinatorial objects of type ${\rm A}^{(1)}_{n-1}$, ${\rm C}^{(1)}_{n-1}$, ${\rm A}^{(2)}_{2n-2}$, ${\rm D}^{(2)}_{n}$

本文针对四类经典仿射李代数对应的量子包络代数不可约可积最高权模晶体基，借助扩展杨图、杨墙给出其多面体实现的显式形式，并给出$B(\infty)$上$\varepsilon_k^*$函数的组合描述。

$B(λ)$ 与 $B(\infty)$ 的一个新的 Young 墙实现

A new Young wall realization of $B(λ)$ and $B(\infty)$

本文利用 Young 墙的新组合学，为量子仿射代数构造了任意水平最高权晶体 $B(λ)$ 和 $B(\infty)$ 的新实现，并证明了约化 Young 墙晶体与 $B(λ)$ 同构。

A uniform realization of the combinatorial $R$-matrix

Kirillov-Reshetikhin crystals are colored directed graphs encoding the structure of certain finite-dimensional representations of affine Lie algebras. A tensor products of column shape Kirillov-Reshetikhin crystals has recently been realized in a uniform way, for all untwisted affine types, in terms of the quantum alcove model. We enhance this model by using it to give a uniform realization of the combinatorial $R$-matrix, i.e., the unique affine crystal isomorphism permuting factors in a tensor product of KR crystals. In other words, we are generalizing to all Lie types Schützenberger's slidi…

A Tensor Product Theorem Related To Perfect Crystals

Kang et al. provided a path realization of the crystal graph of a highest weight module over a quantum affine algebra, as certain semi-infinite tensor products of a single perfect crystal. In this paper, this result is generalized to give a realization of the tensor product of several highest weight modules. The underlying building blocks of the paths are finite tensor products of several perfect crystals. The motivation for this work is an interpretation of fermionic formulas, which arise from the combinatorics of Bethe Ansatz studies of solvable lattice models, as branching functions of affi…

Geometric construction of crystal bases for quantum generalized Kac-Moody algebras

We provide a geometric realization of the crystal $B(\infty)$ for quantum generalized Kac-Moody algebras in terms of the irreducible components of certain Lagrangian subvarieties in the representation spaces of a quiver.

晶格基的多面体实现与经典仿射类型的Young墙

Polyhedral realizations for crystal bases and Young walls of classical affine types

本文研究了经典仿射类型晶格基的多面体实现，通过Young墙和截断墙描述了晶格基的多面体和锥体的不等式，给出了ε_k*-函数的组合描述。

Polyhedral realizations for $B(\infty)$ and extended Young diagrams, Young walls of type ${\rm A}^{(1)}_{n-1}$, ${\rm C}^{(1)}_{n-1}$, ${\rm A}^{(2)}_{2n-2}$, ${\rm D}^{(2)}_{n}$

The crystal bases are quite useful combinatorial tools to study the representations of quantized universal enveloping algebras $U_q(\mathfrak{g})$. The polyhedral realization for $B(\infty)$ is a combinatorial description of the crystal base, which is defined as an image of embedding $Ψ_ι:B(\infty)\hookrightarrow \mathbb{Z}^{\infty}_ι$, where $ι$ is an infinite sequence of indices and $\mathbb{Z}^{\infty}_ι$ is an infinite $\mathbb{Z}$-lattice with a crystal structure associated with $ι$. It is a natural problem to find an explicit form of the polyhedral realization ${\rm Im}(Ψ_ι)$. In this ar…

Crystal graphs for basic representations of the quantum affine algebra $U_q(C_2^{(1)})$

We give a realization of crystal graphs for basic representations of the quantum affine algebra $U_q(C_2^{(1)})$ in terms of new combinatorial objects called the Young walls.