论文标题
HOPF代数$ k^{s_ {n-1}}} \＃kc_n $的指示公式
An Indicator Formula for the Hopf Algebra $k^{S_{n-1}}\#kC_n$
论文作者
论文摘要
对于代数封闭的字段$ k $，使用$ C_N $和$ s_n $和$ s_ {n-1} $的匹配的配对操作来构建了半杂音bismash产品Hopf代数$ j_n = k^{s_ {n-1}} \＃kc_n $。在这项工作中，我们重新解释了这些动作，并利用$ s_ {n-1} $的不适当的理解，以推导$ j_n $的iRREPS的新的froebnius-schur指示器公式，并证明对于$ n $奇数，所有$ j_n $的指标均为无效。我们还得出了包括定理6.2.2在内的各种计数公式，其中充分描述了所有$ 2 $ 2 $维的指标的$ j_n $和定理6.1.2的指标，该指标充分描述了$ j_n $的所有奇数irrepers的指标，并使用这些非零指标表示了$ n $ n $ n $ n $ n $ n $ n $ n。
The semisimple bismash product Hopf algebra $J_n=k^{S_{n-1}}\#kC_n$ for an algebraically closed field $k$ is constructed using the matched pair actions of $C_n$ and $S_{n-1}$ on each other. In this work, we reinterpret these actions and use an understanding of the involutions of $S_{n-1}$ to derive a new Froebnius-Schur indicator formula for irreps of $J_n$ and show that for $n$ odd, all indicators of $J_n$ are nonnegative. We also derive a variety of counting formulas including Theorem 6.2.2 which fully describes the indicators of all $2$-dimensional irreps of $J_n$ and Theorem 6.1.2 which fully describes the indicators of all odd-dimensional irreps of $J_n$ and use these formulas to show that nonzero indicators become rare for large $n$.
