论文标题
关于两个量子仿期代数的两个简单模块的张量产物的简单性
On the simplicity of the tensor product of two simple modules of quantum affine algebras
论文作者
论文摘要
Lapid和Mínguez给出了抛物线感应$σ\ timesπ$的不可约性的标准，其中$σ$是梯子的表示，$π$是对非Archimedean领域的通用线性组的任意不可减至的表示。 Through quantum affine Schur-Weyl duality, when $k$ is large enough, this gives a criterion of the irreducibility of the tensor product of a snake module $L(M)$ and any simple module $L(N)$ of the quantum affine algebra $U_q(\widehat{\mathfrak{sl}_k})$.本文的目的是将条件添加到其标准中，以便适用于任何$ k \ geq 1 $。在两个模块都是蛇模块的情况下，我们证明了标准，或者一个是末端节点的基本模块，而另一个是任何简单的模块。我们还定义了Grasmmannian群集代数$ \ Mathbb {C} [\ Mathrm {gr}（k，n，\ sim）] $的类似标准，并证明对于任何$ k \ geq 1 $，两个梯子，两个梯子仅兼容，并且仅在相应的tableaux满足条件时才兼容。这概括了Leclerc和Zelevinsky的结果，即两个Plücker坐标是兼容的，并且仅当它们弱分离时。
Lapid and Mínguez gave a criterion of the irreducibility of the parabolic induction $σ\times π$, where $σ$ is a ladder representation and $π$ is an arbitrary irreducible representation of the general linear group over a non-archimedean field. Through quantum affine Schur-Weyl duality, when $k$ is large enough, this gives a criterion of the irreducibility of the tensor product of a snake module $L(M)$ and any simple module $L(N)$ of the quantum affine algebra $U_q(\widehat{\mathfrak{sl}_k})$. The goal of this paper is to add conditions to their criterion such that it works for any $k \geq 1$. We prove the criterion in the case where both modules are snake modules or one of them is a fundamental module at an extremity node and the other is any simple module. We also defined a similar criterion in the Grassmannian cluster algebra $\mathbb{C}[\mathrm{Gr}(k,n, \sim)]$, and show that for any $k \geq 1$, two ladders are compatible if and only if the corresponding tableaux satisfy the criterion. This generalizes Leclerc and Zelevinsky's result that two Plücker coordinates are compatible if and only if they are weakly separated.
