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arXiv 1911.03220math.RT
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关于Young置换模的对称幂和外幂

On the symmetric and exterior powers of Young permutation modules

Yu Jiang

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39 pages
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摘要：

设$n$是正整数，$λ$是$n$的一个划分。设$M^λ$是标记为$λ$的Young置换模。本文研究了正特征情形下$M^λ$的对称幂和外幂.我们确定的对称和外部权力的$M^λ$是投射。对$M^λ$的所有不可分解的外幂也进行了分类。然后，我们证明了一些结果的不可分解的直接和项，具有最大的复杂性，在直和分解的一些对称和外部权力的$M^λ$。最后，我们参数化所有的斯科特模块同构的对称或外平方$M^λ$的直接和，并确定其相应的多重性明确。

英文摘要：

Let $n$ be a positive integer and $λ$ be a partition of $n$. Let $M^λ$ be the Young permutation module labelled by $λ$. In this paper, we study symmetric and exterior powers of $M^λ$ in positive characteristic case. We determine the symmetric and exterior powers of $M^λ$ that are projective. All the indecomposable exterior powers of $M^λ$ are also classified. We then prove some results for indecomposable direct summands that have the largest complexity in direct sum decompositions of some symmetric and exterior powers of $M^λ$. We end by parameterizing all the Scott modules that are isomorphic to direct summands of the symmetric or exterior square of $M^λ$ and determining their corresponding multiplicities explicitly.

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