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.
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…
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 realizations of crystal bases. We also observe certain relations of decorations and monomial realizations of crystal bases.
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.
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 realizations of crystal bases.
This is an exposition of the results on Geometric crystals and the associated Kashiwara crystal bases (presented by the first author in RIMS, August 2004)
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 irreducible crystals, which is, in general, different from the decomposition for tensor product of the fundamental crystals.
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 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…
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.
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.
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…
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.
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.
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.
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.
We calculate the image of the combinatorial R-matrix for any classical highest weight element in the tensor product of Kirillov--Reshetikhin crystals $B^{r,k}\otimes B^{1,l}$ of type $D^{(1)}_n, B^{(1)}_n, A^{(2)}_{2n-1}$. The notion of $\pm$-diagrams is effectively used for the identification of classical highest weight elements in $B^{1,l}\otimes B^{r,k}$.
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.
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.
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.
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.
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.
Polyhedral realizations for crystal bases of integrable highest weight modules and combinatorial objects of type ${\rm A}^{(1)}_{n-1}$, ${\rm C}^{(1)}_{n-1}$, ${\rm A}^{(2)}_{2n-2}$, ${\rm D}^{(2)}_{n}$
Let $B(Λ_0)$ be the level 1 highest weight crystal of the quantum affine algebra $U_q(A_n^{(1)})$. We construct an explicit crystal isomorphism between the geometric realization $\mathbb{B}(Λ_0)$ of $B(Λ_0)$ via quiver varieties and the path realization ${\mathcal P}^{\rm ad}(Λ_0)$ of $B(Λ_0)$ arising from the adjoint crystal $\adjoint$.
For any polynomial representation of the special linear group, the nodes of the corresponding crystal may be indexed by semi-standard Young tableaux. Under certain conditions, the standard Young tableaux occur, and do so with weight 0. Standard Young tableaux also parametrize the vertices of dual equivalence graphs. Motivated by the underlying representation theory, in this paper, we explainthis connection by giving a combinatorial manifestation of Schur-Weyl duality. In particular, we put a dual equivalence graph structure on the 0-weight space of certain crystal graphs, producing edges combi…
We study the polytope model for the affine type $A$ Kirillov-Reshetikhin crystals and prove that the action of the affine Kashiwara operators can be described in a remarkable simple way. Moreover, we investigate the combinatorial $R$-matrix on a tensor product of polytopes and characterize the map explicitly on the highest weight elements. We further give a formula for the local energy function and provide an alternative proof for the perfectness. We determine for any dominant highest weight element $Λ$ of level $\ell$ the elements $b_Λ, b^Λ$ involved in the definition of perfect crystals and …
The main object in this paper is a certain rational convex polytope whose lattice points give a polyhedral realization of a highest weight crystal basis. This is also identical to a Newton-Okounkov body of a flag variety, and it gives a toric degeneration. In this paper, we prove that a specific class of this polytope is given by Kiritchenko's Demazure operators on polytopes. This implies that polytopes in this class are all lattice polytopes. As an application, we give a sufficient condition for the corresponding toric variety to be Gorenstein Fano.
The tableau model for Kirillov-Reshetikhin (KR) crystals, which are finite dimensional crystals corresponding to certain affine Lie algebras, is commonly used for its ease of crystal operator calculations. However, its simplicity makes quite complex the calculation of statistics such as: keys (used to express Demazure characters), the crystal energy function (an affine grading on tensor products of KR crystals), and the combinatorial R-matrix (an affine crystal isomorphism permuting factors in a tensor product of KR crystals). It has been shown that these calculations are much simpler with the…
This is a companion paper to arXiv:0909.2280. It is mostly expository and focuses on the representation-theoretic and combinatorial aspects of the main problems considered in the other article.
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 $\mathcal{B}(λ)$.
Let $B(Λ)$ be a level $\ell$ highest weight crystal of the quantum affine algebra $U_q(A_n^{(1)})$. We construct an explicit crystal isomorphism between the geometric realization $\gB(Λ)$ of the crystal $B(Λ)$ using quiver varieties and the path realization $\mathcal{P}^{\rm ad}(Λ)$ arising from the adjoint crystal $\Bad{\ell}$ of level $\ell$.
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…
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…
We present a simple combinatorial model for the characters of the irreducible integrable highest weight modules for complex symmetrizable Kac-Moody algebras. This model can be viewed as a discrete counterpart to the Littelmann path model. We describe crystal graphs and give a Littlewood-Richardson rule for decomposing tensor products of irreducible representations. The new model is based on the notion of a lambda-chain, which is a chain of positive roots defined by certain interlacing conditions.
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…
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…
We give a new realization of crystal bases for finite dimensional irreducible modules over special linear Lie algebras using the monomials introduced by H. Nakajima. We also discuss the connection between this monomial realization and the tableau realization given by Kashiwara and Nakashima.
In this paper, we give an explicit combinatorial realization of the crystal B(λ) for an irreducible highest weight U_q(q(n))-module V(λ) in terms of semistandard decomposition tableaux. We present an insertion scheme for semistandard decomposition tableaux and give algorithms of decomposing the tensor product of q(n)-crystals. Consequently, we obtain explicit combinatorial descriptions of the shifted Littlewood-Richardson coefficients.
The Mullineux map is a combinatorial function on partitions which describes the effect of tensoring a simple module for the symmetric group in characteristic $p$ with the one-dimensional sign representation. It can also be interpreted as an isomorphism between crystal graphs for $\widehat{\mathfrak{sl}}_p$. We give a new combinatorial description of the Mullineux map by expressing this crystal isomorphism as a composition of isomorphisms between different crystals. These isomorphisms are defined in terms of new generalised regularisation maps introduced by Millan Berdasco. We then given two ap…
The rigged configuration realization $RC(\infty)$ of the crystal $B(\infty)$ was originally presented as a certain connected component within a larger crystal. In this work, we make the realization more concrete by identifying the elements of $RC(\infty)$ explicitly for the $A_n$-type case. Two separate descriptions of $RC(\infty)$ are obtained. These lead naturally to isomorphisms $RC(\infty)\cong T(\infty)$ and $RC(\infty)\cong \bar{T}(\infty)$, i.e., those with the marginally large tableau and marginally large reverse tableau realizations of $B(\infty)$, that may be computed explicitly. We …
We discuss Cauchy type decompositions of crystal graphs for general linear Lie superalgebras. More precisely, we consider bicrystal graph structures on various sets of matrices of non-negative integers, and obtain their decompositions with explicit combinatorial isomorphisms.
We construct a combinatorial crystal structure on the Kirillov-Reshetikhin crystal $B^{7,s}$ in type $E_7^{(1)}$, where $7$ is the unique node in the orbit of $0$ in the affine Dynkin diagram. We then describe the combinatorial $R$-matrix $R \colon B^{7,s} \otimes B^{7,s'} \to B^{7,s'} \otimes B^{7,s}$.
We present a list of ``local'' axioms and an explicit combinatorial construction for the regular $B_2$-crystals (crystal graphs of highest weight integrable modules over $U_q(sp_4)$). Also a new combinatorial model for these crystals is developed.
We study properties of the realizations of groups as the combinatorial automorphism group of a convex polytope. We show that for any non-abelian group $G$ with a central involution there is a centrally symmetric polytope with $G$ as its combinatorial automorphisms. We show that for each integer $n$, there are groups that cannot be realized as the combinatorial automorphisms of convex polytopes of dimension at most $n$. We also give an optimal lower bound for the dimension of the realization of a group as the group of isometries that preserves a convex polytope.
A regular $A_n$-crystal is an edge-colored directed graph, with $n$ colors, related to an irreducible highest weight integrable module over $U_q(sl_{n+1})$. Based on Stembridge's local axioms for regular simply-laced crystals and a structural characterization of regular $A_2$-crystals in \cite{DKK-07}, we present a new combinatorial construction, the so-called {\em crossing model}, and prove that this model generates precisely the set of regular $A_n$-crystals. Using the model, we obtain a series of results on the combinatorial structure of such crystals and properties of their subcrystals.
The geometric Satake correspondence provides an equivalence of categories between the Satake category of spherical perverse sheaves on the affine Grassmannian and the category of representations of the dual group. In this note, we define a combinatorial version of the Satake category using irreducible components of fibres of the convolution morphism. We then prove an equivalence of coboundary categories between this combinatorial Satake category and the category of crystals of the dual group.
For simply-laced Kac-Moody algebras $\frak g$, Stembridge (2003) proposed a `local' axiomatization of crystal graphs of representations of $U_q(\frak g)$. In this paper we propose axioms for edge-2-colored graphs which characterize the crystals of integrable representations of $U_q(sp(4))$, regular crystal graphs of $B_2$-type. An edge-colored directed graph which obeys our Axioms (K0)--(K5) is called an R-{\em graph} (for brevity), and our main result is that the regular crystals of $B_2$-type are R-graphs and vice versa. We give a direct combinatorial construction for the crystals in questio…
We give an algebraic description of several modules and algebras related to the vector partition function, and we prove that they can be realized as the equivariant K-theory of some manifolds that have a nice combinatorial description. We also propose a more natural and general notion of duality between these modules, which corresponds to a Poincaré duality-type correspondence for equivariant K-theory.
