论文标题
量子格拉斯曼尼亚人及其相关的量子舒伯特品种在团结根源
Quantum Grassmannians and their Associated Quantum Schubert Varieties at roots of unity
论文作者
论文摘要
我们研究了统一根部各种量子代数的PI度，包括量子拉格曼尼亚人，量子舒伯特品种，分区亚代代代代数及其相关的量子仿射空间。通过de concini和procesi的定理，分区亚词法的PI及其相关的量子仿射空间由与（Cauchon-le）图相关的偏斜 - 对称积分矩阵控制。我们证明，这些矩阵的不变因素始终是2的功率。这使我们能够明确计算分区子代理的PI程度。 我们的结果还适用于某些完全质量（均质）的分区亚词法词。特别是，我们的结果使我们能够扩展jakobsen和Jondrup关于统一根部[JJ01]根部量子确定环的PI程度的结果，并且我们提出了一种构建最大维度的不可减至表示量子确定性理想的最大维度的方法。 在这些结果的基础上，我们通过非交通性除剂量[LR08]使用分区亚代构和量子舒伯特品种之间的牢固联系来获得量子schubert品种的PI程度的表达式。特别是，我们计算了量子拉格曼人的PI程度。
We study the PI degree of various quantum algebras at roots of unity, including quantum Grassmannians, quantum Schubert varieties, partition subalgebras, and their associated quantum affine spaces. By a theorem of De Concini and Procesi, the PI degree of partition subalgebras and their associated quantum affine spaces is controlled by skew-symmetric integral matrices associated to (Cauchon-Le) diagrams. We prove that the invariant factors of these matrices are always powers of 2. This allows us to compute explicitly the PI degree of partition subalgebras. Our results also apply to certain completely prime (homogeneous) quotients of partition subalgebras. In particular, our results allow us to extend results of Jakobsen and Jondrup regarding the PI degree of quantum determinantal rings at roots of unity [JJ01] and we present a method to construct an irreducible representation of maximal dimension for quantum determinantal ideals. Building on these results, we use the strong connection between partition subalgebras and quantum Schubert varieties through noncommutative dehomogenisation [LR08] to obtain expressions for the PI degree of quantum Schubert varieties. In particular, we compute the PI degree of quantum Grassmannians.
