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A Clebsch-Gordan decomposition in positive characteristic
Journal of Algebra ( IF 0.8 ) Pub Date : 2020-10-01 , DOI: 10.1016/j.jalgebra.2020.06.001
Stephen Donkin , Samuel Martin

Let $G$ be the special linear group of degree $2$ over an algebraically closed field $K$. Let $E$ be the natural module and $S^rE$ the $r$th symmetric power. We consider here, for $r,s\geq 0$, the tensor product of $S^rE$ and the dual of $S^sE$. In characteristic zero this tensor product decomposes according to the Clebsch-Gordan formula. We consider here the situation when $K$ is a field of positive characteristic. We show that each indecomposable component occurs with multiplicity one and identify which modules occur for given $r$ and $s$.

中文翻译:

具有正特征的 Clebsch-Gordan 分解

令 $G$ 是代数闭域 $K$ 上度数 $2$ 的特殊线性群。令 $E$ 为自然模,$S^rE$ 为第 $r$ 个对称幂。我们在这里考虑,对于 $r,s\geq 0$,$S^rE$ 的张量积和 $S^sE$ 的对偶。在特征零中,该张量积根据 Clebsch-Gordan 公式分解。我们在这里考虑 $K$ 是一个正特性场的情况。我们展示了每个不可分解的组件以多重性出现并确定哪些模块出现在给定的 $r$ 和 $s$ 中。
更新日期:2020-10-01
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