sun123zxy's blog

sun123zxy's blog https://blog.sun123zxy.top/listings/all/ All posts hosted on this site. https://blog.sun123zxy.top/asterisk.png sun123zxy's blog https://blog.sun123zxy.top/listings/all/ 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 Lustig–Spaltenstein：分块幂零轨道的提升 sun123zxy https://blog.sun123zxy.top/posts/20260409-ncone-induced-orbit/ 中的 Lustig–Spaltenstein 提升：使用分块对角幂零轨道拼出整个 $\mathfrak{gl}_n$ 中的幂零轨道． ]]> math algebra lie https://blog.sun123zxy.top/posts/20260409-ncone-induced-orbit/ Fri, 10 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 作用群的选择 sun123zxy https://blog.sun123zxy.top/posts/20260330-ncone-ad-group/ 上篇中简单介绍了复数域上 $\mathfrak{gl}_n$ 或 $\mathfrak{sl}_n$ 的共轭 / 幂零轨道的定义．这是 ad hoc 的．一般来说，不同的群作用在李代数 $\mathfrak{g}$ 上可能导致不同的轨道．本篇介绍我们选择作用群的标准，主要参考 [1, section 1.2]． ]]> math algebra lie https://blog.sun123zxy.top/posts/20260330-ncone-ad-group/ Mon, 30 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 On the Rank and the Span Rank of Modules sun123zxy https://blog.sun123zxy.top/posts/20251118-rank/ math algebra commalg https://blog.sun123zxy.top/posts/20251118-rank/ Tue, 18 Nov 2025 00:00:00 GMT