what is a supercharacter
play

What is... a supercharacter? Dario De Stavola 11 October 2016 What - PowerPoint PPT Presentation

Dec 22, 2023 •262 likes •1.24k views

What is... a supercharacter? What is... a supercharacter? Dario De Stavola 11 October 2016 What is... a supercharacter? Basic representation theory G finite group, V finite dimensional C - vector space of dimension d What is... a

  1. What is... a supercharacter? What is... a supercharacter? Dario De Stavola 11 October 2016

  2. What is... a supercharacter? Basic representation theory G finite group, V finite dimensional C - vector space of dimension d

  3. What is... a supercharacter? Basic representation theory G finite group, V finite dimensional C - vector space of dimension d X : G → GL ( V ) is a C -linear representation.

  4. What is... a supercharacter? Basic representation theory G finite group, V finite dimensional C - vector space of dimension d X : G → GL ( V ) is a C -linear representation. Fix a basis for V ∼ = C d , then X : G → GL d ( C ).

  5. What is... a supercharacter? Basic representation theory Examples The trivial representation: ∀ g ∈ G , X ( g ) = 1 d ;

  6. What is... a supercharacter? Basic representation theory Examples The trivial representation: ∀ g ∈ G , X ( g ) = 1 d ; G = Z / 4 Z = { 0 , 1 , 2 , 3 } , V = C , GL ( V ) = C × , then

  7. What is... a supercharacter? Basic representation theory Examples The trivial representation: ∀ g ∈ G , X ( g ) = 1 d ; G = Z / 4 Z = { 0 , 1 , 2 , 3 } , V = C , GL ( V ) = C × , then 1 = X (0) = X (1 + 1 + 1 + 1) = X (1) 4

  8. What is... a supercharacter? Basic representation theory Examples The trivial representation: ∀ g ∈ G , X ( g ) = 1 d ; G = Z / 4 Z = { 0 , 1 , 2 , 3 } , V = C , GL ( V ) = C × , then 1 = X (0) = X (1 + 1 + 1 + 1) = X (1) 4 X (1) ∈ { 1 , i , − 1 , − i }

  9. What is... a supercharacter? Basic representation theory Examples The trivial representation: ∀ g ∈ G , X ( g ) = 1 d ; G = Z / 4 Z = { 0 , 1 , 2 , 3 } , V = C , GL ( V ) = C × , then 1 = X (0) = X (1 + 1 + 1 + 1) = X (1) 4 X (1) ∈ { 1 , i , − 1 , − i } X (1) = i , X (2) = − 1 , X (3) = − i , X (0) = 1 .

  10. What is... a supercharacter? Basic representation theory Examples G = Z / 2 Z × Z / 2 Z , X : G → GL 2 ( C )

  11. What is... a supercharacter? Basic representation theory Examples G = Z / 2 Z × Z / 2 Z , X : G → GL 2 ( C ) � � � � 1 0 0 − i X (0 , 0) = , X (1 , 0) = , 0 1 i 0 � � � � − 1 0 0 i X (0 , 1) = , X (1 , 1) = . 0 − 1 − i 0

  12. What is... a supercharacter? Basic representation theory Irreducibles X : G → GL d ( C ) , X ( g ): V → V

  13. What is... a supercharacter? Basic representation theory Irreducibles X : G → GL d ( C ) , X ( g ): V → V W ≤ V ⇒ X ( g ) | W : W → V

  14. What is... a supercharacter? Basic representation theory Irreducibles X : G → GL d ( C ) , X ( g ): V → V W ≤ V ⇒ X ( g ) | W : W → V If X ( g ) | W : W → W ⇒ X W : G → GL ( W ) is a subrepresentation

  15. What is... a supercharacter? Basic representation theory Irreducibles X : G → GL d ( C ) , X ( g ): V → V W ≤ V ⇒ X ( g ) | W : W → V If X ( g ) | W : W → W ⇒ X W : G → GL ( W ) is a subrepresentation if W ≤ V ⇒ V = W ⊕ U

  16. What is... a supercharacter? Basic representation theory Irreducibles X : G → GL d ( C ) , X ( g ): V → V W ≤ V ⇒ X ( g ) | W : W → V If X ( g ) | W : W → W ⇒ X W : G → GL ( W ) is a subrepresentation if W ≤ V ⇒ V = W ⊕ U Manschke’s theorem U is also a subrepresentation.

  17. What is... a supercharacter? Basic representation theory Irreducibles V = W ⊕ U , W , U subrepresentations

  18. What is... a supercharacter? Basic representation theory Irreducibles V = W ⊕ U , W , U subrepresentations X ( g )( v ) = X ( g )( w + u ) = X W ( g )( w ) ⊕ X U ( g )( u )

  19. What is... a supercharacter? Basic representation theory Irreducibles V = W ⊕ U , W , U subrepresentations X ( g )( v ) = X ( g )( w + u ) = X W ( g )( w ) ⊕ X U ( g )( u ) � � X W ( g ) 0 X ( g ) ∼ = 0 X U ( g )

  20. What is... a supercharacter? Basic representation theory Irreducibles V = W ⊕ U , W , U subrepresentations X ( g )( v ) = X ( g )( w + u ) = X W ( g )( w ) ⊕ X U ( g )( u ) � � X W ( g ) 0 X ( g ) ∼ = 0 X U ( g ) We need to study only representations without nontrivial subrepresentations! These representations are called Irreducible .

  21. What is... a supercharacter? Basic representation theory Irreducibles V = W ⊕ U , W , U subrepresentations X ( g )( v ) = X ( g )( w + u ) = X W ( g )( w ) ⊕ X U ( g )( u ) � � X W ( g ) 0 X ( g ) ∼ = 0 X U ( g ) We need to study only representations without nontrivial subrepresentations! These representations are called Irreducible . The number of irreducible representations is equal to the number of conjugacy classes of G

  22. What is... a supercharacter? Basic representation theory Characters X : G → GL d ( C ) The character χ : G → C is defined as χ ( g ) := tr ( X ( g )).

  23. What is... a supercharacter? Basic representation theory Characters X : G → GL d ( C ) The character χ : G → C is defined as χ ( g ) := tr ( X ( g )). � � � � 1 0 0 − i X (0 , 0) = , X (1 , 0) = , 0 1 i 0 � � � � − 1 0 0 i X (0 , 1) = , X (1 , 1) = . 0 − 1 − i 0

  24. What is... a supercharacter? Basic representation theory Characters X : G → GL d ( C ) The character χ : G → C is defined as χ ( g ) := tr ( X ( g )). � � � � 1 0 0 − i X (0 , 0) = , X (1 , 0) = , 0 1 i 0 � � � � − 1 0 0 i X (0 , 1) = , X (1 , 1) = . 0 − 1 − i 0 χ (0 , 0) = 2 , χ (1 , 0) = 0 , χ (0 , 1) = − 2 , χ (1 , 1) = 0 .

  25. What is... a supercharacter? Basic representation theory Properties χ (1 G ) = d ,

  26. What is... a supercharacter? Basic representation theory Properties χ (1 G ) = d , called the degree of the representation;

  27. What is... a supercharacter? Basic representation theory Properties χ (1 G ) = d , called the degree of the representation; χ ( g − 1 ) = χ ( g );

  28. What is... a supercharacter? Basic representation theory Properties χ (1 G ) = d , called the degree of the representation; χ ( g − 1 ) = χ ( g ); χ ( h − 1 gh ) = χ ( g ),

  29. What is... a supercharacter? Basic representation theory Properties χ (1 G ) = d , called the degree of the representation; χ ( g − 1 ) = χ ( g ); χ ( h − 1 gh ) = χ ( g ), the character is a class function .

  30. What is... a supercharacter? Basic representation theory Properties χ (1 G ) = d , called the degree of the representation; χ ( g − 1 ) = χ ( g ); χ ( h − 1 gh ) = χ ( g ), the character is a class function . A character is irreducible if it is the trace of an irreducible representation.

  31. What is... a supercharacter? Basic representation theory Frobenius scalar product Let φ, ψ : G → C

  32. What is... a supercharacter? Basic representation theory Frobenius scalar product Let φ, ψ : G → C � φ, ψ � := 1 � φ ( g ) ψ ( g ) ∈ C | G | g ∈ G

  33. What is... a supercharacter? Basic representation theory Frobenius scalar product Let φ, ψ : G → C � φ, ψ � := 1 � φ ( g ) ψ ( g ) ∈ C | G | g ∈ G Irreducible characters are orthonormal w.r.t. this product: if χ 1 , χ 2 are irreducible characters then � χ 1 , χ 2 � = δ { χ 1 = χ 2 }

  34. What is... a supercharacter? Basic representation theory Class functions ∼ : conjugation (we say g ∼ g ′ if g ′ = h − 1 gh )

  35. What is... a supercharacter? Basic representation theory Class functions ∼ : conjugation (we say g ∼ g ′ if g ′ = h − 1 gh ) Cl G := { f : G → C class function } ∼ = { f : G / ∼ → C } = ( G / ∼ ) ∗

  36. What is... a supercharacter? Basic representation theory Class functions ∼ : conjugation (we say g ∼ g ′ if g ′ = h − 1 gh ) Cl G := { f : G → C class function } ∼ = { f : G / ∼ → C } = ( G / ∼ ) ∗ Since | G / ∼ | = | Irr ( G ) | , and irreducible characters are othonormal w.r.t �· , ·�

  37. What is... a supercharacter? Basic representation theory Class functions ∼ : conjugation (we say g ∼ g ′ if g ′ = h − 1 gh ) Cl G := { f : G → C class function } ∼ = { f : G / ∼ → C } = ( G / ∼ ) ∗ Since | G / ∼ | = | Irr ( G ) | , and irreducible characters are othonormal w.r.t �· , ·� Irr ( G ) is an orthonormal basis for Cl G .

  38. What is... a supercharacter? Basic representation theory Character table · · · K 1 K 2 K 3 χ 1 χ 1 ( K 1 ) χ 1 ( K 2 ) χ 1 ( K 3 ) χ 2 χ 2 ( K 1 ) χ 2 ( K 2 ) χ 2 ( K 3 ) χ 3 χ 3 ( K 1 ) χ 3 ( K 2 ) χ 3 ( K 3 ) . . . . . .

  39. What is... a supercharacter? Basic representation theory Character table · · · K 1 K 2 K 3 χ 1 χ 1 ( K 1 ) χ 1 ( K 2 ) χ 1 ( K 3 ) χ 2 χ 2 ( K 1 ) χ 2 ( K 2 ) χ 2 ( K 3 ) χ 3 χ 3 ( K 1 ) χ 3 ( K 2 ) χ 3 ( K 3 ) . . . . . . This table is orthonormal!

  40. What is... a supercharacter? Basic representation theory Example G = Z / 3 Z

  41. What is... a supercharacter? Basic representation theory Example G = Z / 3 Z 0 1 2 χ 1 1 1 1 2 π i π i χ 2 1 e e 3 3 2 π i π i χ 3 1 e e 3 3

  42. What is... a supercharacter? Basic representation theory X : G → GL ( V ) , Y : G → GL ( V ) , χ = tr ( X ) , γ = tr ( Y )

  43. What is... a supercharacter? Basic representation theory X : G → GL ( V ) , Y : G → GL ( V ) , χ = tr ( X ) , γ = tr ( Y ) X ∼ = Y ⇔ χ = γ

  44. What is... a supercharacter? Basic representation theory X ∼ = Y ⇔ χ = γ proof [ ⇒ ] X ∼ ⇒ = Y

More recommend