sun123zxy's blog

sun123zxy's blog https://blog.sun123zxy.top/listings/short/ Short expository notes to capture insights. https://blog.sun123zxy.top/alice-gtm.jpg sun123zxy's blog https://blog.sun123zxy.top/listings/short/ quarto-1.9.34 Mon, 20 Apr 2026 00:00:00 GMT $\Delta \implies \otimes$ Gemini 3 Flash https://blog.sun123zxy.top/posts/20260420-alg-rep-tensor/ the tensor product of two representations of a generic associative algebra is not naturally a representation. ]]> math algebra https://blog.sun123zxy.top/posts/20260420-alg-rep-tensor/ Mon, 20 Apr 2026 00:00:00 GMT An Intrinsic, Boilerplate-Free Definition of Affine Varieties sun123zxy https://blog.sun123zxy.top/posts/20260331-intrinsic-affine-variety/ [1, definition 2.1.6] gives an intrinsic, boilerplate-free definition of affine varieties as follows (note our “variety” is not necessarily irreducible): ]]> math algebra https://blog.sun123zxy.top/posts/20260331-intrinsic-affine-variety/ Wed, 01 Apr 2026 00:00:00 GMT Lie 定理 sun123zxy https://blog.sun123zxy.top/posts/20260331-lie-engel/ [1, Sec. 1.2]． ]]> math algebra lie https://blog.sun123zxy.top/posts/20260331-lie-engel/ Tue, 31 Mar 2026 00:00:00 GMT 置换速算技巧 sun123zxy https://blog.sun123zxy.top/posts/20260326-cycle-notation-tech/ ： ]]> math algebra https://blog.sun123zxy.top/posts/20260326-cycle-notation-tech/ Thu, 26 Mar 2026 00:00:00 GMT 三句话证明迹的循环不变性 sun123zxy https://blog.sun123zxy.top/posts/20260314-trace-cyclic/ day，我们三句话为迹的循环不变性 $\operatorname{Tr}(ABC) = \operatorname{Tr}(BCA) = \operatorname{Tr}(CAB)$ 提供一种圆润的理解．接受这一理解的前置条件是掌握自然同构 $\operatorname{Hom}(V,W) \cong W \otimes V^* \cong V^* \otimes W$ ． ]]> math algebra https://blog.sun123zxy.top/posts/20260314-trace-cyclic/ Sat, 14 Mar 2026 00:00:00 GMT Spec，可约与连通性 sun123zxy https://blog.sun123zxy.top/posts/20260228-spec-connect/ math algebra commalg https://blog.sun123zxy.top/posts/20260228-spec-connect/ Fri, 27 Feb 2026 00:00:00 GMT Comparing Free Modules via Homomorphisms sun123zxy https://blog.sun123zxy.top/posts/20260225-free-module-cmp/ be a nonzero commutative ring with $1$ . ]]> math algebra commalg https://blog.sun123zxy.top/posts/20260225-free-module-cmp/ Wed, 25 Feb 2026 00:00:00 GMT 透镜成像乱炖 sun123zxy https://blog.sun123zxy.top/posts/20251218-lens/ 这里 $n_1, n_2$ 分别是两介质的折射率， $\theta_1, \theta_2$ 分别是入射角和折射角． ]]> math physics https://blog.sun123zxy.top/posts/20251218-lens/ Thu, 18 Dec 2025 00:00:00 GMT Factorials and Integer Partitions sun123zxy https://blog.sun123zxy.top/posts/20250615-partorial/ objects from $n$ distinct objects, we have binomial coefficients defined as $\binom n k := \frac{n!}{k!(n-k)!}$ A slightly more general case is when we want to partition $n$ distinct objects into several labeled sets, with the size of each set is identified by an integer partition $(\lambda_1,\lambda_2,\dots) = \lambda \vdash n$ . In such circumstance, the multinomial coefficients, $\binom{n}{\lambda} := \frac{n!}{\lambda!} := \frac{n!}{\prod_i \lambda_i!}$ counts the number of ways to do so. So here comes the first variant $\lambda! := \prod_i \lambda_i!$ ]]> math combinatorics https://blog.sun123zxy.top/posts/20250615-partorial/ Sun, 15 Jun 2025 00:00:00 GMT 诱导表示乱炖 sun123zxy https://blog.sun123zxy.top/posts/20250606-indrep/ [1] ]]> math algebra group-rep https://blog.sun123zxy.top/posts/20250606-indrep/ Sat, 07 Jun 2025 00:00:00 GMT 推广的 Cayley-Hamilton 定理及其应用 sun123zxy https://blog.sun123zxy.top/posts/20250604-cayley-hamilton/ 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 $\varphi$ 在某组基下的矩阵表示．我们将这一定理稍做推广： ]]> math algebra https://blog.sun123zxy.top/posts/20250604-cayley-hamilton/ Wed, 04 Jun 2025 00:00:00 GMT 有限群表示的 Maschke 定理是分裂模正合列的提升 sun123zxy https://blog.sun123zxy.top/posts/20250525-maschke/ 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 $0 \to U \xrightarrow{\iota} V \to V/U \to 0$ 已经知道，上列作为 $K$ -模正合列分裂，可设其有左分裂 $K$ -模同态 $\pi: V \to U$ ．我们希望把这一分裂同态提升到 $K[G]$ -模结构上．构造 $\begin{aligned} \tilde \pi : V &\to U \\ v &\mapsto \frac 1 {|G|} \sum_{g \in G} g \cdot \pi (g^{-1} \cdot v) \end{aligned}$ 验证它是 $K[G]$ -模同态同时仍然满足 $\tilde \pi \circ \iota = \mathrm{id}_U$ 即可．这样就得到 $K[G]$ -模同构 $V = U \oplus \ker \tilde \pi$ ． ]]> math algebra https://blog.sun123zxy.top/posts/20250525-maschke/ Sun, 25 May 2025 00:00:00 GMT