---
created: 2022-02-23T18:45
updated: 2023-03-31T16:01
title: representation theory
aliases: ["representation theory", "Representation Theory", "representations", "representation", "Levi"]
---

---

- Tags 
	- #lie-theory #MOC 
- Refs:
	- Basic linear algebra: <http://perso.ens-lyon.fr/sophie.morel/MAT_217_notes.pdf#page=1> #resources/course-notes 
	- Basic homological algebra: <http://perso.ens-lyon.fr/sophie.morel/notes540.pdf#page=5> #resources/course-notes 
	- <https://services.math.duke.edu/~jdr/papers/building.pdf> #resources/papers 
	- Ernest B. Vinberg: Linear Representations of Groups #resources/books 
	- Reps of finite groups: <https://www2.math.upenn.edu/~zvihr/courses/RTNotes.pdf#page=15> #resources/course-notes 
	- Lots of notes <https://ivganev.github.io/notes/>
	- Princetone course notes: <http://perso.ens-lyon.fr/sophie.morel/rep_theory_notes.pdf#page=7>
	- Reps of topological groups: <http://perso.ens-lyon.fr/sophie.morel/notes_449.pdf#page=5> #resources/course-notes 
- Links:
	- [algebraic group](Unsorted/algebraic%20group.md)
	- [induction](induction.md)
	- [depth](depth.md)
	- [parahoric](Unsorted/parahoric.md)
	- [parabolic](Unsorted/parabolic.md)
	- [Weyl group](Unsorted/Weyl%20group.md)
	- [Unsorted/Kazhdan Lusztig conjecture](Unsorted/Kazhdan%20Lusztig%20conjecture.md)
	- [Unsorted/BGG resolution](Unsorted/BGG%20resolution.md)
	- [Pontryagin dual](Unsorted/Pontryagin%20dual.md)
	- [perverse sheaf](Unsorted/perverse%20sheaf.md)

---

# representation theory

# Groups

- [Cartan](Cartan)
- [Levi](Levi)
- [Borel](Borel.md)
- [maximal torus](maximal%20torus)
- [split torus](split%20torus)
- [Regular representation](Regular%20representation)
- [Maschke's theorem](Maschke's%20theorem.md)
- [Schur's lemma](Schur's%20lemma.md)

# Algebraic Groups

- [algebraic group](algebraic%20group.md)
- [Hopf algebra](Hopf%20algebra.md)
- [unipotent radical](unipotent%20radical)
- [Lie algebra](Lie%20algebra.md)
- [Verma module](Verma%20module)
- [Category O](Category%20O)
- [Wall-crossing morphisms](Wall-crossing%20morphisms)

# Groups

- Basic definitions and properties of representations, including [Schur's Lemma](Schur's%20Lemma.md) and [Maschke's Theorem](Maschke's%20Theorem.md).
- The representation theory of finite groups, including [Schur orthogonality](Schur%20orthogonality).
- Fundamental constructions such as tensor product, dual representations and induced representations.
- Representation theory of compact groups, including the [Peter-Weyl Theorem](Peter-Weyl%20Theorem).
- Description of the [irreducible representations](irreducible%20representations) of $S_n, \SU_2, \SO_3, \liesl_2(\CC)$.

# Notation

![](attachments/2023-03-31-13.png)

# Notes

![](attachments/2023-03-31-14.png)
![](attachments/2023-03-31-15.png)
![](attachments/2023-03-31-16.png)
![](attachments/2023-03-31-17.png)
![](attachments/2023-03-31-18.png)

## Smooth and admissible reps
![](attachments/2023-03-31-19.png)

![](attachments/2023-03-31-20.png)

## Supercuspidal reps

![](attachments/2023-03-31-21.png)
![](attachments/2023-03-31-22.png)

## Classification

![](attachments/2023-03-31-23.png)
![](attachments/2023-03-31-24.png)

# Topics

## Basics
- [simple](simple)
	- [semisimple](semisimple)
- [irreducibles](irreducibles)
- [indecomposable objects of a category](indecomposable%20objects%20of%20a%20category)
- [classification of representations of compact Lie groups](classification%20of%20representations%20of%20compact%20Lie%20groups)
- [characters](characters.md)
- [weight of a representation](weight%20of%20a%20representation)
- [Frobenius reciprocity](Frobenius%20reciprocity)

## Finite Groups
- [Maschke's theorem](Maschke's%20theorem.md)
	- [completely reducible](completely%20reducible)
- [Schur's Lemma](Schur's%20Lemma.md)
- [sign representation](sign%20representation)
- [permutation representation](permutation%20representation)

## Lie Groups

- [root system](root%20system.md)
- [Weyl group](Weyl%20group.md)
	- [Coxeter groups](Coxeter%20groups)
	- [Bruhat order](Bruhat%20order)

## Advanced

- [Verma module](Verma%20module)
- [Category O](Category%20O)
- [Hecke algebra](Hecke%20algebra)
	- [categorification](categorification.md)
- [Engel's theorem](Engel's%20theorem)


Algebras, representations, Schur's lemma. Representations of SL(2). Representations of finite groups, Maschke's theorem, characters, applications. Induced representations, Burnside's theorem, Mackey formula, Frobenius reciprocity. Representations of quivers.

# Results

An irreducible representation of $G$ is completely determined by its character.


# Levis

![](2023-03-31-69.png)
![](2023-03-31-70.png)