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Factorials and Integer Partitions

math
combinatorics
Factorials are commonly seen in combinatorics, and today we talk about some variants of them related to integer partitions. Recall that if one wish to count the number of…
2025/06/15
en
short
 

推广的 Cayley-Hamilton 定理及其应用

math
algebra
线性代数中的 Cayley-Hamilton 定理指出,域 \(K\) 上线性变换 \(\varphi\) 的特征多项式 \(f(\lambda) = \det(\lambda I_n - \Phi)\) 是它的一个零化多项式,这里 \(\Phi \in \operatorname{Mat}_{n \times n}(K)\) 是 \(\varp…
2025/06/04
zh
short
 

有限群表示的 Maschke 定理是分裂模正合列的提升

math
algebra
所谓 Maschke 定理,是指有限群表示的半单(可完全分解)性.可以考虑如下正合列风格的理解.设 \(G\) 是有限群,\(K\) 是特征不为 \(|G|\) 的域,\(K[G]\) 是 \(G\) 的群代数;\(V\) 是 \(K[G]\)-模,\(U\) 为 \(V\) 的子模.我们有 \(K[G]\)-模正合列 \[ 0 \to U…
2025/05/25
zh
short
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