On the structure of cyclotomic nilHecke algebras
On the structure of cyclotomic nilHecke algebras
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DOI：
10.2140/pjm.2018.296.105
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发表时间：
2017-09
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影响因子：
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通讯作者：
Jun Hu;Xinfeng Liang
中科院分区：
文献类型：
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作者：
Jun Hu;Xinfeng Liang
本文研究了环切nilHecke代数$\HH_{\ell,n}^{(0)}$的结构，其中$\ell,n\ In \ n $。我们构造了$\HH_{\ well,n}^{(0)}$的单项式基来验证Mathas的一个猜想。证明了$\HH_{\ well,n}^{(0)}$的梯度基本代数是可交换的，因此与$\HH_{\ well,n}^{(0)}$的中心$Z$同构。进一步证明$\HH_{\嗯，n}^{(0)}$与$Z$上的满矩阵代数同构，并构造了中心$Z$的显式基。构造了$\HH_{\ well,n}^{(0)}$的一对正交本元幂等完备集。最后，通过在$\HH_{\ell,n}^{(0)}$的给定齐次基上显式指定$\Tr$的值，给出了$\HH_{\ell,n}^{(0)}$上新的齐次对称形式$\Tr$，并证明了它与Shan—Varagnolo—Vasserot在$\HH_{\ell,n}^{(0)}$上的对称形式$\Tr^{\text{SVV}}$一致。
In this paper we study the structure of the cyclotomic nilHecke algebras $\HH_{\ell,n}^{(0)}$, where $\ell,n\in\N$. We construct a monomial basis for $\HH_{\ell,n}^{(0)}$ which verifies a conjecture of Mathas. We show that the graded basic algebra of $\HH_{\ell,n}^{(0)}$ is commutative and hence isomorphic to the center $Z$ of $\HH_{\ell,n}^{(0)}$. We further prove that $\HH_{\ell,n}^{(0)}$ is isomorphic to the full matrix algebra over $Z$ and construct an explicit basis for the center $Z$. We also construct a complete set of pairwise orthogonal primitive idempotents of $\HH_{\ell,n}^{(0)}$. Finally, we present a new homogeneous symmetrizing form $\Tr$ on $\HH_{\ell,n}^{(0)}$ by explicitly specifying its values on a given homogeneous basis of $\HH_{\ell,n}^{(0)}$ and show that it coincides with Shan--Varagnolo--Vasserot's symmetrizing form $\Tr^{\text{SVV}}$ on $\HH_{\ell,n}^{(0)}$.