Quantum affine algebras and Grassmannians
Quantum affine algebras and Grassmannians
复制标题
量子仿射代数和格拉斯曼代数
DOI：
10.1007/s00209-020-02496-7
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发表时间：
2020-02-27
影响因子：
0.8
通讯作者：
Li, Jian-Rong
中科院分区：
文献类型：
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作者：
Chang, Wen;Duan, Bing;Li, Jian-Rong
研究了A型量子仿射代数与Grassmannian簇代数之间的关系。埃尔南德斯和勒克莱尔描述了一个同构从Grothendieck环的某个子范畴Cl的Uq（sln）-模的商的格拉斯曼集群代数中的某些冻结变量设置为1。我们解释了这是如何诱导的幺半群的占主导地位的单项式，用于parametriums简单的模块，和一个商的幺半群的矩形半标准杨tableaux与n行和条目在[n +1 + 1]。通过同构，我们定义了Grassmannian簇代数中每个矩形表T的元素ch（T）。根据Kashiwara，Kim，Oh和Park以及Qin的结果，每个Grassmannian丛单项式对于某个T都是ch（T）形式的.利用Arakawa-Suzuki公式，给出了ch（T）的一个显式表达式，并给出了有限维Uq（sln）-模的一个显式q-特征标公式.我们给出了一个在格拉斯曼簇代数中进行突变的表论规则。我们建议如何使用我们的公式来研究现实和素的模块，和集群变量的兼容性。
We study the relation between quantum affine algebras of type A and Grassmannian cluster algebras. Hernandez and Leclerc described an isomorphism from the Grothendieck ring of a certain subcategory Cl of Uq(sln)-modules to a quotient of the Grassmannian cluster algebra in which certain frozen variables are set to 1. We explain how this induces an isomorphism between the monoid of dominant monomials, used to parameterize simple modules, and a quotient of the monoid of rectangular semistandard Young tableaux with n rows and with entries in [n+l+1]. Via the isomorphism, we define an element ch(T) in a Grassmannian cluster algebra for every rectangular tableau T. By results of Kashiwara, Kim, Oh, and Park, and also of Qin, every Grassmannian cluster monomial is of the form ch(T) for some T. Using a formula of Arakawa-Suzuki, we give an explicit expression for ch(T), and also give explicit q-character formulas for finite-dimensional Uq(sln)-modules. We give a tableau-theoretic rule for performing mutations in Grassmannian cluster algebras. We suggest how our formulas might be used to study reality and primeness of modules, and compatibility of cluster variables.