量子对称对的编织与有限维表示论
Webs and finite dimensional Representation theory for quantum symmetric pairs
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中文总结 AI 辅助
本文为量子对称对构造三角分解与Verma模，定义图解范畴$Web^B$控制多项式表示，证明量子外幂无重数分解并确定不可约和项。
中文摘要 AI 辅助
我们考虑类型AIII（拟分裂）量子对称对子代数$U_q'$。我们在两个子族之一中定义了一个新的Cartan子代数。与另一个子族中Letzter的Cartan子代数一起，我们通过Letzter映射构造了$U_q'$的三角分解。然后我们将Verma模定义为诱导模，并证明有限维单$U_q'$-模是我们Verma模的商。这些处理在两个子族中是一致的。在第二部分中，我们定义了一个图解范畴$Web^B$，它控制$U_q'$的多项式表示。我们证明了量子外幂$\bigwedge^k\mathbb{V}$作为$U_q'$-模具有无重数分解，使用了$Web^B$中某些点态射的显式特征空间分解。作为应用，我们推导了所涉及的反对称球Hecke模的无重数性质。对于$\bigwedge^k\mathbb{V}$，我们显式确定了最高权向量及其权，从而确定了其不可约直和项的同构类。最后，我们确定了$Web^B$作用在$U_q'$的某个有限维表示范畴上的核。
英文摘要
We consider the type AIII (quasi split) quantum symmetric pair subalgebras $U_q'$. We define a new Cartan subalgebra in one of the two subfamilies. Together with Letzter's Cartan subalgebra in the other, we construct a triangular decomposition of $U_q'$ via the Letzter map. Then we define Verma modules as induced modules and prove that finite dimensional simple $U_q'$-modules are quotients of our Verma modules. These treatments are uniform in both subfamilies. In the second part we define a diagrammatic category $Web^B$ that controls polynomial representations of $U_q'$. We prove a multiplicity-free decomposition of the quantum exterior powers $\bigwedge^k\mathbb{V}$ as a $U_q'$-module, using a explicit eigenspace decomposition of certain dot morphisms in $Web^B$. As applications, we deduce the multiplicity-freeness of the involved anti-spherical Hecke module. For $\bigwedge^k\mathbb{V}$, we explicitly determine the highest weight vectors and their weights, thereby determining the isomorphism classes of their irreducible summands. Finally, we determine the kernel of the $Web^B$-action on certain finite dimensional representation category of $U_q'$.
发表机构
- University of Bonn（波恩大学）
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