[Notes] Representation of Finite and Compact Groups

Textbook: Representations of Finite and Compact Groups, Simon

這是康明昌老師的開課用書，確實如他所述、這本書的問題很多，常常會有符號上的錯誤，也不適合第一次唸群表現的人讀。

但是這本書由很高的角度來看群表現這件事，多半的入門書會花很多精力在處理實例(譬如說對稱群)，一個接一個介紹那些 representation，但不系統，雖然說這本書也沒有很系統的把這些講好，但是它提供了一個觀點試圖這樣作(待我好好整理整理...)

之後待補上 Sagan 的群表現筆記，那個比較適合第一次學的人看。

Notes
Different forms of a representation:
group homomorphism
group action
matrix representation
module
unitary map
Adjoint/unitary operator
Polar Decomposition
Irreducible representation/Completely Reducible
Direct Sum
Maschke Theorem
Group Algebra $A(G)$ and its representation
Convolution
$A(G) = < $ irreps 
Schur Lemma
Tensor Product and its representation
Conjugation/conjugate representation/Anti unitary map
Self-conjugate representation
real/quaternion/complex
Quaternion Algebra

Notes
Examples:
(Finite) Abelian groups
cyclic groups
$S3$
$D4$
$H$ (quaternion group)
Character
irrep $\leftrightarrow$ inner product of character = 1
Orthogonal relation
Wedderburn Theorem
infinite dimension compact Lie group
Center of Group Algebra
Irreps of G $\leftrightarrow$ conjugacy classes of G
Orthogonality relation
Abelianization
Examples:
    Character table of $A4$
Ito Theorem
Frobenius-Schur Indicator
Class Function
Ambivalent conjugacy class $\leftrightarrow$ self-conjugate irreps
Burnside Theorem/ $p^a q^b$ Theorem
Group Determinant
Irreps over $H$

Notes
Irreps over $\mathbb{R}$
rep over $R$ $\leftrightarrow$ rep over $\mathcal{C}$ with complex conjugation
complex conjugation/anti-unitary map over $H$ and $R$
Tensor Product Rep. over $H$ and $R$
Restriction of irrep
Induced Representation
example of index 2
Sufficient and necessary condition of an induced rep is an irrep
restriction/induced between $S_4$ and $A_4$
When an induced rep. is real/quaternion/complex
$S_4$'s character table induced by $S_3$
Clifford Group $D(n)$

Notes
Clifford Group fundamental properties
$Z(D(n))$
square of generator
irreps
characters
When $D(n)$ and $D^+(n)$ are real/quaternion/complex
Clifford Theory
Wigner-Mackey's method of little groups
Semi-direct Product
Eigenspace and its orbits/stabilizer
Examples: $D_n$, $A_4$, $S_4$

Notes
Coinduced Module
Frobenius Character Formula
Frobenius Reciprocity Theorem (Example: $S_5$)
character table of $A_5$
Double Coset Decomposition
Mackey Theorem I,II,III
group of Lie type: $PSL_2(q)$ and its Borel subgroup

Notes
Symmetric Group (Revised)
Frame, Young's (standard) tableux
Branching Relation (dimension)
Enumeration of Standard Tableux by Residue Theorem
Projections in $A(S_n)$
Row/Column Group
Minimal Idempotent Element/Minimal Left Ideal
Central Idempotent Element
Branching Relations
Specht Module
Frobenius/Schur Character Formula