Part 4 群的线性表示的结构

Math Online Tom Circle

不变子空间: Invariant Sub-space

第一课: Direct Sum 直和 $latex oplus$of Representations

直和 = $latex {oplus}&fg=aa0000&s=3$

第二课: 群表示可约 Reducible Representation

Analogy :
Prime number decomposition
Irreducible Polynomial

外直和 : $latex { dot{ +} }&fg=aa0000&s=3$

$latex boxed { displaystyle phi_{1} dot {+} phi_{2} = tilde {phi_{1}} oplus tilde {phi_{2}}}&fg=aa0000&s=3$

* 第三课: 完全可约表示 Completely Reducible Representation

完全表示是可 完全分解为 不可约表示 的一种表示。

完全可约表示 => 其子表示 也 完全可约
不可约 一定是完全可约的!
[Analogy: Polynomial degree 1 (x + 1) is irreducible. ]

註: (*) 深奥课, 可以越过直接跳到结果。(证明 待以后 复习)。

集合证明: 交(和)和(交)

如果 也是⊆ , 则 交(和) =和(交)
Ref 2 《高代》 Pg 250 命题 1

$latex boxed {U cap (U_{1} oplus W) supseteq (U cap U_{1} ) oplus (U cap W)}&fg=aa0000&s=3$
$latex U cap (U_{1} oplus W) subseteq (U cap U_{1} ) oplus (U cap W)$
$latex boxed {U…

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