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sun123zxy's blog https://blog.sun123zxy.top/listings/math/ Mathematical posts. https://blog.sun123zxy.top/kourindou.jpg sun123zxy's blog https://blog.sun123zxy.top/listings/math/ 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 梦话集 sun123zxy https://blog.sun123zxy.top/posts/20260419-rambling/ math https://blog.sun123zxy.top/posts/20260419-rambling/ Sun, 19 Apr 2026 00:00:00 GMT Schur–Weyl 对偶 sun123zxy https://blog.sun123zxy.top/posts/20260419-schur-weyl/ 是群，则对任意 $G$ 的 $n$ 维表示 $V$ ， $V \otimes_{\mathbb C} V$ 都可以分解为子表示 $\operatorname{Sym}^2 V$ 和 $\operatorname{Alt}^2 V$ 的直和，其中 ]]> math algebra https://blog.sun123zxy.top/posts/20260419-schur-weyl/ Sun, 19 Apr 2026 00:00:00 GMT Jacobson–Morozov：$\mathfrak{sl}_2$ 在复半单李代数中的嵌入 sun123zxy https://blog.sun123zxy.top/posts/20260410-jacobson-morozov/ 中工作，并假设抽象 Jordan 分解、Killing 型和 Cartan 根空间分解理论已建立完毕．这一部分的标准参考是 [1, section 3.2–3.3]． ]]> math algebra lie https://blog.sun123zxy.top/posts/20260410-jacobson-morozov/ Fri, 10 Apr 2026 00:00:00 GMT DVR–Dedekind 乱炖 sun123zxy https://blog.sun123zxy.top/posts/20260406-dvr-dedekind/ 上篇中对零维 Noether 环的结构做了详尽的分析，局部情形的主要结果如下： ]]> math algebra commalg https://blog.sun123zxy.top/posts/20260406-dvr-dedekind/ Tue, 07 Apr 2026 00:00:00 GMT 幂零轨道的维数 sun123zxy https://blog.sun123zxy.top/posts/20260322-ncone-dim/ 上篇介绍了幂零锥和幂零轨道的基本概念．本篇我们关心 $\mathfrak{gl}_n$ 幂零锥和幂零轨道的维数情况． ]]> math algebra lie https://blog.sun123zxy.top/posts/20260322-ncone-dim/ Fri, 03 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 $\mathfrak{gl}_n$，幂零轨道与支配序 sun123zxy https://blog.sun123zxy.top/posts/20260321-geo-nilp-gln/ 或 $\mathfrak{sl}_n$ 切入，入门友好地为李代数上幂零锥的几何提供一些感觉． ]]> math algebra lie https://blog.sun123zxy.top/posts/20260321-geo-nilp-gln/ Sat, 21 Mar 2026 00:00:00 GMT 有限群表示论速通 sun123zxy https://blog.sun123zxy.top/posts/20260312-group-rep-speedrun/ 是有限群， $V$ 是（有限维的） $\mathbb C$ -线性空间，则群同态 $\rho: G \to \operatorname{GL}(V)$ 规定了一个 $G$ 的（复）表示．等价地，这为 $V$ 配备了一个 $G$ -模结构—— $\mathbb C[G]$ -模结构的简写． ]]> math algebra https://blog.sun123zxy.top/posts/20260312-group-rep-speedrun/ Sun, 15 Mar 2026 00:00:00 GMT 有限群表示论：Peter–Weyl 定理 sun123zxy https://blog.sun123zxy.top/posts/20260314-group-rep-peter-weyl/ 这里 $\mathbb C[G]$ 是群 $G$ 的群代数， $V_i$ 是 $G$ 的全体不可约表示．这只是个左 $\mathbb C[G]$ -模的分解： $\mathbb C[G]$ 被分解成了其单左理想的直和．但是 $\mathbb C[G]$ 是个环，完整来说我们应该研究其作为 $\mathbb C$ -代数的分解．Peter–Weyl / Wedderburn–Artin 定理给出了它的分解： $\mathbb C[G] \cong \bigoplus_i \operatorname{End}_{\mathbb C}(V_i) \cong \bigoplus_i V_i \otimes_{\mathbb C} V_i^*$ 可见其结构确实变得更加丰富．稍做辨析： ]]> math algebra https://blog.sun123zxy.top/posts/20260314-group-rep-peter-weyl/ Sat, 14 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 Selected solutions to Atiyah-Macdonald’s exercises sun123zxy https://blog.sun123zxy.top/posts/20251218-atiyah/ Exercise 1 ([1]-exr-2.1) $\mathbb Z / m \mathbb Z \otimes_{\mathbb Z} \mathbb Z / n \mathbb Z = 0$ when $m, n$ are coprime. ]]> math algebra commalg https://blog.sun123zxy.top/posts/20251218-atiyah/ Thu, 18 Dec 2025 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 代数批如何理解 Einstein 求和约定 sun123zxy https://blog.sun123zxy.top/posts/20251028-einstein-sum/ Einstein 求和约定（Einstein summation convention）就似乎深受其害：物理学家们习惯于使用 Einstein 求和约定来简化张量矩阵运算中的求和符号，和他们纠缠不清的微分几何学家们也纷纷采用这一记号，并在教学时自豪的向学生介绍这一记号对求和的简化作用．一眼看上去，在 Einstein 求和约定下，你不再显式地写出求和符号和求和指标：每个乘法中左侧的下标和右侧的上标自动配对作为求和指标．例如，两个向量 $\boldsymbol a$ 和 $\boldsymbol b$ 的内积将被写作 $a_i b^i$ ．然而，大部分教学实践中仅仅将 Einstein 求和约定理解为一种“省略求和符号的记号简化”，忽视其背后的代数结构，并不能说服困惑的学生接受这一记号．纯坐标化的理解也往往掩盖了 Einstein 求和约定的内在含义——这是将抽象线性空间张量积和其在基底下的坐标矩阵运算联系起来的桥梁． ]]> math algebra geometry physics https://blog.sun123zxy.top/posts/20251028-einstein-sum/ Thu, 06 Nov 2025 00:00:00 GMT 分析力学乱炖 sun123zxy https://blog.sun123zxy.top/posts/20250921-mechanics/ Newton 第二定律 $m \ddot{\vb* q} = -\pdv{U}{\vb* q}$ 这里 $U(\vb* q)$ 是质点在位置 $\vb* q$ 处的势能．物理学实践似乎表明这种形式下的力学方程长相往往比较丑陋，不易于守恒量的发现．考虑到光学中光路传播的最短路径原理具有优美统一的数学形式，物理学家们开始尝试将力学方程也写成某种变分原理的形式．这种尝试推动了 Lagrange 力学和 Hamilton 力学的发展，也最终将经典力学、光学和量子力学统一在同一框架下． ]]> math analysis physics https://blog.sun123zxy.top/posts/20250921-mechanics/ Sun, 21 Sep 2025 00:00:00 GMT 零点定理乱炖 sun123zxy https://blog.sun123zxy.top/posts/20250709-hnt/ math algebra commalg https://blog.sun123zxy.top/posts/20250709-hnt/ Tue, 12 Aug 2025 00:00:00 GMT