零级权形状的Lakshmibai-Seshadri路径和与零级基本表示相关的一维和

Crystals of Lakshmibai-Seshadri paths and extremal weight modules over quantum hyperbolic Kac-Moody algebras of rank 2

Let $\mathfrak{g}$ be a hyperbolic Kac-Moody algebra of rank $2$, and let $λ$ be an arbitrary integral weight. We denote by $\mathbb{B}(λ)$ the crystal of all Lakshmibai-Seshadri paths of shape $λ$. Let $V(λ)$ be the extremal weight module of extremal weight $λ$ generated by the (cyclic) extremal weight vector $v_λ$ of weight $λ$, and let $\mathcal{B}(λ)$ be the crystal basis of $V(λ)$ with $u_λ\in \mathcal{B}(λ)$ the element corresponding to $v_λ$. We prove that the connected component $\mathcal{B}_0(λ)$ of $\mathcal{B}(λ)$ containing $u_λ$ is isomorphic, as a crystal, to the connected compon…

Quantum Lakshmibai-Seshadri paths and the specialization of Macdonald polynomials at $t=0$ in type $A_{2n}^{(2)}$

In this paper, we give a combinatorial realization of the crystal basis of a quantum Weyl module over a quantum affine algebra of type $A_{2n}^{(2)}$, and a representation-theoretic interpretation of the specialization $P_λ^{A_{2n}^{(2)}} (q,0)$ of the symmetric Macdonald polynomial $P_λ^{A_{2n}^{(2)}} (q,t)$ at $t=0$, where $λ$ is a dominant weight and $P_λ^{A_{2n}^{(2)}}(q,t)$ denotes the specific specialization of the symmetric Macdonald-Koornwinder polynomial $P_λ(q,t_1, t_2, t_3, t_4, t_5)$. More precisely, as some results for untwisted affine types, the set of all ($A_{2n}^{(2)}$-type) q…

Chevalley formula for anti-dominant weights in the equivariant $K$-theory of semi-infinite flag manifolds

We prove a Pieri-Chevalley formula for anti-dominant weights and also a Monk formula in the torus-equivariant $K$-group of the formal power series model of semi-infinite flag manifolds, both of which are described explicitly in terms of semi-infinite Lakshmibai-Seshadri paths (or, equivalently, quantum Lakshmibai-Seshadri paths). In view of recent results of Kato, these formulas give an explicit description of the structure constants for the Pontryagin product in the torus-equivariant $K$-group of affine Grassmannians and that for the quantum multiplication of the torus-equivariant (small) qua…

仿射李代数零级形状的Lakshmibai-Seshadri路集的晶体结构

Crystal structure of the set of Lakshmibai-Seshadri paths of a level-zero shape for an affine Lie algebra

令$λ= \sum_{i \in I_{0}} m_{i} \varpi_{i}$，其中$m_{i} \in \mathbb{Z}_{\ge 0}$为$i \in I_{0}$，是$\mathbb{Q}$上仿射李代数$\mathfrak{g}$的零级优势积分权，其中$\varpi_{i}$，$i \in I_{0}$是零级基本权，并且令$\mathbb{B}(λ)$是所有形状为$λ$的Lakshmibai-Seshadri路的晶体。首先，我们给出了一个明确的描述分解的晶体$\mathbb{B}(λ)$成一个不相交的联盟的连接组件，并表明，所有的连接组件是成对的“同构”（移动的重量）。第二，我们“实现”$\mathbb{B}(λ)$的连通分量$\mathbb{B}_{0}(λ)$，其中包含直线$π_λ$作为$\mathbb{B}(λ)_{\mathrm{cl}} \simeq \bigotimes_{i \in I_{0}} \bigl(\mathbb{B}(\varpi_{i})_{\mathrm{cl}} \bigr)^{\otimes m_{i}}$晶体（具有权格$P_{\mathrm{cl}} = P/(\mathbb{Q}δ\cap P)$，其中$δ$是$\mathfrak{g}$的零根）的仿射$\hat{\mathbb{B}(λ)_{\mathrm{cl}}}$（具有权格…

Demazure submodules of level-zero extremal weight modules and specializations of Macdonald polynomials

In this paper, we give a characterization of the crystal bases $\mathcal{B}_{x}^{+}(λ)$, $x \in W_{\mathrm{af}}$, of Demazure submodules $V_{x}^{+}(λ)$, $x \in W_{\mathrm{af}}$, of a level-zero extremal weight module $V(λ)$ over a quantum affine algebra $U_{q}$, where $λ$ is an arbitrary level-zero dominant integral weight, and $W_{\mathrm{af}}$ denotes the affine Weyl group. This characterization is given in terms of the initial direction of a semi-infinite Lakshmibai-Seshadri path, and is established under a suitably normalized isomorphism between the crystal basis $\mathcal{B}(λ)$ of the le…

秩 $2$ Kac-Moody 代数基本表示的单项式实现与 LS 路径

Monomial realizations and LS paths of fundamental representations for rank $2$ Kac-Moody algebras

本文针对秩 $2$ Kac-Moody 代数，显式构造了基本权的 Lakshmibai-Seshadri 路径集合与单项式实现之间的同构，并基于 Weyl 群给出了单项式实现的显式形式。

Specialization of nonsymmetric Macdonald polynomials at $t=\infty$ and Demazure submodules of level-zero extremal weight modules

In this paper, we give a representation-theoretic interpretation of the specialization $E_{w_{\circ} λ} (q,\infty)$ of the nonsymmetric Macdonald polynomial $E_{w_{\circ} λ}(q,t)$ at $t=\infty$ in terms of the Demazure submodule $V_{w_\circ}^{-} (λ)$ of the level-zero extremal weight module $V(λ)$ over a quantum affine algebra of an arbitrary untwisted type, here, $λ$ is a dominant integral weight, and $w_{\circ}$ denotes the longest element in the finite Weyl group $W$. Also, for each $x \in W$, we obtain a combinatorial formula for the specialization $E_{x λ} (q, \infty)$ at $t=\infty$ of th…

Path model for an extremal weight module over the quantized hyperbolic Kac-Moody algebra of rank 2

Let $\mathfrak{g}$ be a hyperbolic Kac-Moody algebra of rank 2, and set $λ=Λ_{1} - Λ_{2}$, where $Λ_{1}$, $Λ_{2}$ are the fundamental weights. Denote by $V(λ)$ the extremal weight module of extremal weight $λ$ with $v_λ$ the extremal weight vector, and by $\mathcal{B}(λ)$ the crystal basis of $V(λ)$ with $u_λ$ the element corresponding to $v_λ$. We prove that (i) $\mathcal{B}(λ)$ is connected, (ii) the subset $\mathcal{B}(λ)_μ$ of elements of weight $μ$ in $\mathcal{B}(λ)$ is a finite set for every integral weight $μ$, and $\mathcal{B}(λ)_λ = \{u_λ\}$, (iii) every extremal element in $\mathcal…

Tableau models for semi-infinite Bruhat order and level-zero representations of quantum affine algebras

We prove that semi-infinite Bruhat order on an affine Weyl group is completely determined from those on the quotients by affine Weyl subgroups associated with various maximal (standard) parabolic subgroups of finite type. Furthermore, for an affine Weyl group of classical type, we give a complete classification of all cover relations of semi-infinite Bruhat order (or equivalently, all edges of the quantum Bruhat graphs) on the quotients in terms of tableaux. Combining these we obtain a tableau criterion for semi-infinite Bruhat order on an affine Weyl group of classical type. As an application…