Dong-Uy Shin
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.
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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.
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.
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 $\mathcal{B}(λ)$.
We study the monomial crystal defined by the second author. We show that each component of the monomial crystal can be embedded into a crystal of an extremal weight module introduced by Kashiwara. And we determine all monomials appearing in the components corresponding to all level 0 fundamental representations of quantum affine algebras except for some nodes of $E_6^{(2)}$, $E_7^{(1)}$, $E_8^{(1)}$. Thus we obtain explicit descriptions of the crystals in these examples. We also give those for the corresponding finite dimensional representations. For classical types, we give them in terms of t…
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 construct a new family of representations for the quantum toroidal algebras of type $A_n$, which are $\ell$-extremal in the sense of Hernandez [24]. We construct extremal loop weight modules associated to level 0 fundamental weights $\varpi_\ell$ when $n=2r+1$ is odd and $\ell=1, r+1$ or $n$. To do it, we relate monomial realizations of level 0 extremal fundamental weight crystals with integrable representations of $\mathcal{U}_q(sl_{n+1}^{tor})$, and we introduce promotion operators for the level 0 extremal fundamental weight crystals. By specializing the quantum parameter, w…
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.
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。
Let $λ$ be a (level-zero) dominant integral weight for an untwisted affine Lie algebra, and let $\mathrm{QLS}(λ)$ denote the quantum Lakshmibai-Seshadri (QLS) paths of shape $λ$. For an element $w$ of a finite Weyl group $W$, the specializations at $t = 0$ and $t = \infty$ of the nonsymmetric Macdonald polynomial $E_{w λ}(q, t)$ are explicitly described in terms of QLS paths of shape $λ$ and the degree function defined on them. Also, for (level-zero) dominant integral weights $λ$, $μ$, we have an isomorphism $Θ: \mathrm{QLS}(λ+ μ) \rightarrow \mathrm{QLS}(λ) \otimes \mathrm{QLS}(μ)$ of crystal…
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 …
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 $…
We study the crystal base of the negative part of a quantum group. Two explicit descriptions of the crystal $B(\infty)$ for types $G_2$ are given. The first is given in terms of extended Nakajima monomials and the second realization follows a similar result given for other finite types by Cliff.
We present a vertex operator algebra which is an extension of the level $k$ vertex operator algebra for the $\hat{sl}_2$ conformal field theory. We construct monomial basis of its irreducible representations.
We investigate the affine canonical basis and the monomial basis constructed in [LXZ] in Lusztig's geometric setting. We show that the transition matrix between the two bases is upper triangular with 1's in the diagonal and coefficients in the upper diagonal entries in N[v, v^{-1}]. As a consequence, we show that part of the monomial basis elements give rise to resolutions of support varieties of the affine canonical basis elements as simple perverse sheaves.
In 1990 Beilinson, Lusztig and MacPherson provided a geometric realization of modified quantum $\mathfrak{gl}_n$ and its canonical basis. A key step of their work is a construction of a monomial basis. Recently, Du and Fu provided an algebraic construction of the canonical basis for modified quantum affine $\mathfrak{gl}_n$, which among other results used an earlier construction of monomial bases using Ringel-Hall algebra of the cyclic quiver. In this paper, we give an elementary algebraic construction of a monomial basis for affine Schur algebras and modified quantum affine $\mathfrak{gl}_n$.
Canonical bases via pairing monomials
本文提出一种基于配对单项式的新公式与Lusztig-Shoji算法结合，用于计算有限ADE型量子群的典范基，并在A型中推广至计算仿射Hecke代数标准模的合成重数及简单模维数。
We provide $\mathbb{N}$-filtrations on the negative part $U_q(\mathfrak{n}^-)$ of the quantum group associated to a finite-dimensional simple Lie algebra $\mathfrak{g}$, such that the associated graded algebra is a skew-polynomial algebra on $\mathfrak{n}^-$. The filtration is obtained by assigning degrees to Lusztig's quantum PBW root vectors. The possible degrees can be described as lattice points in certain polyhedral cones. In the classical limit, such a degree induces an $\mathbb{N}$-filtration on any finite dimensional simple $\mathfrak{g}$-module. We prove for type $\tt{A}_n$, $\tt{C}_n…
Monomial bases and canonical bases for quantum affine algebras
本文研究了量子仿射代数中单项基与规范基的构造方法，提出了一种基于PBW基的简单算法来计算规范基，并探讨了由此得到的规范基与Lusztig利用有向图几何构造的规范基之间的关系。
We construct a monomial basis of the positive part of the quantized enveloping algebra associated to a finite-dimensional simple Lie algebra. As an application we give a simple proof of the existence and uniqueness of the canonical basis of the positive part.
Applications of the freezing operators on cluster algebras
本研究将冻结算子应用于（量子）上cluster代数，证明其保持局部化（量子）cluster单项式与基的映射性质，且冻结构造的基与局部化构造的基相一致，为不同cluster代数的关联提供了算子方法。
We give a systematic description of many monomial bases for a given quantized enveloping algebra and of many integral monomial bases for the associated Lusztig $\mathbb Z[v,v^{-1}]$-form. The relations between monomial bases, PBW bases and canonical bases are also discussed.
PBW bases and marginally large tableaux in type D
本文刻画D型$B(\infty)$的边缘大表实现与基于最长词良好约化表达式的PBW单项式实现之间的唯一晶体同构。
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…
We use the idea of generic extensions to investigate the correspondence between the isomorphism classes of nilpotent representations of a cyclic quiver and the orbits in the corresponding representation varieties. We endow the set $\cal M$ of such isoclasses with a monoid structure and identify the submonoid $\cal M_c$ generated by simple modules. On the other hand, we use the partial ordering on the orbits (i.e., the Bruhat-Chevalley type ordering) to induce a poset structure on $\cal M$ and describe the poset ideals generated by an element of the submonoid $\cal M_c$ in terms of the existenc…
Feigin-Stoyanovsky's type subspaces for affine Lie algebras of type $C_\ell^{(1)}$ have monomial bases with a nice combinatorial description. We describe bases of whole standard modules in terms of semi-infinite monomials obtained as "a limit of translations" of bases for Feigin-Stoyanovsky's type subspaces.
The Haar State values of monomials and a method to pick orthonormal bases on $O(U_q(3))$
本文研究量子群 $O(U_q(n))$ 上单项式的 Haar 态求值问题，在 $O(U_q(3))$ 上给出显式公式，并应用于计算不可约余表示的 Gram 矩阵，为构造标准正交基提供关键工具。
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