THE SECOND FUNDAMENTAL THEOREM OF INVARIANT THEORY FOR THE ORTHOGONAL GROUP
THE SECOND FUNDAMENTAL THEOREM OF INVARIANT THEORY FOR THE ORTHOGONAL GROUP
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DOI：
10.4007/annals.2012.176.3.12
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发表时间：
2011-02
影响因子：
4.9
通讯作者：
G. Lehrer;Rui-bin Zhang
中科院分区：
文献类型：
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作者：
G. Lehrer;Rui-bin Zhang
令V = C n 具有正交形式，G = O(V ) 为相应的正交群。布劳尔 (Brauer) 在 1937 年证明存在满射同态：Br(n)！ EndG(V r )，其中 Br(n) 是参数为 n 的 r 弦布劳尔代数。然而，其核心仍然难以捉摸。在本文中，我们证明，与 GL(V ) 的情况类似，对于 r n + 1，具有由单个幂等元素 E 生成的内核，并且我们给出了 E 的简单显式公式。利用元胞代数理论，我们展示了如何使用 E 来确定 V r 中 O(V ) 的不可约表示的重数。我们还展示了我们的结果如何扩展到 C 被适当的正特征场取代的情况，并对我们结果的量子类似物进行了评论。
Let V = C n be endowed with an orthogonal form and G = O(V ) be the corresponding orthogonal group. Brauer showed in 1937 that there is a surjective homomorphism : Br(n)! EndG(V r ), where Br(n) is the r-string Brauer algebra with parameter n. However the kernel of has remained elusive. In this paper we show that, in analogy with the case of GL(V ), for r n + 1, has a kernel which is generated by a single idempotent element E, and we give a simple explicit formula for E. Using the theory of cellular algebras, we show how E may be used to determine the multiplicities of the irreducible representations of O(V ) in V r . We also show how our results extend to the case where C is replaced by an appropriate eld of positive characteristic, and we comment on quantum analogues of our results.