Modular Representation Theory Study of representations of G over k when char( k ) = p Every representation can be written as a direct sum of indecomposable representations.
Modular Representation Theory Study of representations of G over k when char( k ) = p Every representation can be written as a direct sum of indecomposable representations. Understand the indecomposable representations
Representations of Cyclic p -groups ◮ G cyclic of order p n
Representations of Cyclic p -groups ◮ G cyclic of order p n ◮ The representation theory is “nice”
Representations of Cyclic p -groups ◮ G cyclic of order p n ◮ The representation theory is “nice” ◮ Finitely many indecomposable representations, know how to describe them
Representations of Cyclic p -groups ◮ G cyclic of order p n ◮ The representation theory is “nice” ◮ Finitely many indecomposable representations, know how to describe them ◮ Same is true for groups containing cyclic p -groups
Lemma Let G be a cyclic p-group, | G | = p n , and let k be a field of characteristic p. The only irreducible representation of G is the trivial representation.
Lemma Let G be a cyclic p-group, | G | = p n , and let k be a field of characteristic p. The only irreducible representation of G is the trivial representation. #( irreducible representations) = #( conjugacy classes of elements g ∈ G with p not dividing order of g )
Lemma Let G be a cyclic p-group, | G | = p n , and let k be a field of characteristic p. The only irreducible representation of G is the trivial representation. #( irreducible representations) = #( conjugacy classes of elements g ∈ G with p not dividing order of g ) kG decomposes into a sum of the projective indecomposable representations = k [ x ] / ( x p n − 1) ∼ kG ∼ = k [ x ] / ( x − 1) p n (( a − b ) p = a p − b p since char( k ) = p ) one projective indecomposable representation ⇒ one irreducible representation
Indecomposable Representations | G | = p n , G = � σ � Trivial g �→ 1 for all g ∈ G , matrix (1) Representation � 1 � Dimension 2 1 σ �→ Representation 0 1 1 1 0 Dimension 3 σ �→ 0 1 1 Representation 0 0 1 . . . . . . 1 1 · · · 0 0 0 1 0 0 Dimension p n . . ... ... . . σ �→ . . Representation 0 0 · · · 1 1 0 0 · · · 0 1
Example: Cyclic Group Consider G = Z / 3 Z = { 0 , 1 , 2 } , σ = 1, char( k ) = 3
Example: Cyclic Group Consider G = Z / 3 Z = { 0 , 1 , 2 } , σ = 1, char( k ) = 3 Indecomposable Representations 1 1 0 � 1 � 1 σ �→ (1) σ �→ σ �→ 0 1 1 0 1 0 0 1
Example: Cyclic Group Consider G = Z / 3 Z = { 0 , 1 , 2 } , σ = 1, char( k ) = 3 Indecomposable Representations 1 1 0 � 1 � 1 σ �→ (1) σ �→ σ �→ 0 1 1 0 1 0 0 1 Compare with....
Example: Cyclic Group Consider G = Z / 3 Z = { 0 , 1 , 2 } , σ = 1, char( k ) = 3 Indecomposable Representations 1 1 0 � 1 � 1 σ �→ (1) σ �→ σ �→ 0 1 1 0 1 0 0 1 Compare with.... char( k ) = 0, Irreducible Representations σ �→ ω σ �→ 1 σ �→ ω 2
Example: Cyclic Group and “Other” Group Consider G = Z / 3 Z ⋊ Z / 4 Z , char( k ) = 3
Example: Cyclic Group and “Other” Group Consider G = Z / 3 Z ⋊ Z / 4 Z , char( k ) = 3 Let ω be a 4 th root of unity in k . We get 3 · 4 = 12 indecomposable representations visualized as follows:
Example: Cyclic Group and “Other” Group Consider G = Z / 3 Z ⋊ Z / 4 Z , char( k ) = 3 Let ω be a 4 th root of unity in k . We get 3 · 4 = 12 indecomposable representations visualized as follows: 1 1 0 � 1 � 1 (1) 0 1 1 0 1 0 0 1 ω 1 0 � ω � 1 ( ω ) 0 ω 1 0 ω 0 0 ω ω 2 1 0 � ω 2 � 1 ω 2 ( ω 2 ) 0 1 ω 2 0 ω 2 0 0 ω 3 1 0 � ω 3 � 1 ( ω 3 ) ω 3 0 1 ω 3 0 ω 3 0 0
The previous example generalizes. Let G = P ⋊ C , where P is cyclic of order p n and C is cyclic of order c , with p ∤ c . There are c · p n indecomposable representations of G . ω c − 1 1-dim (1) ( ω ) · · · � � � � � � ω c − 1 1 1 ω 1 1 2-dim · · · ω c − 1 0 1 0 ω 0 . . . ω c − 1 1 1 · · · 0 ω 1 · · · 0 1 0 . . . ... ... . . . 0 . 0 . 0 . p n -dim · · · . . . ... ... . . . . 1 . 1 . 1 ω c − 1 0 · · · 0 1 0 · · · 0 ω 0 · · ·
Algebraic Geometry Study of algebraic curves
Algebraic Geometry Study of algebraic curves Goal: Define a representation of a group using geometry
Algebraic Geometry Study of algebraic curves Goal: Define a representation of a group using geometry ◮ Define an algebraic curve
Algebraic Geometry Study of algebraic curves Goal: Define a representation of a group using geometry ◮ Define an algebraic curve ◮ Define the module of relative differentials
Algebraic Geometry Study of algebraic curves Goal: Define a representation of a group using geometry ◮ Define an algebraic curve ◮ Define the module of relative differentials ◮ Group actions on curves
Algebraic Geometry Study of algebraic curves Goal: Define a representation of a group using geometry ◮ Define an algebraic curve ◮ Define the module of relative differentials ◮ Group actions on curves ◮ Define a representation using geometry
Example: Affine Plane Curve Let f ( x , y ) ∈ k [ x , y ] be an irreducible polynomial. The affine plane curve defined by f is X f = { ( a , b ) ∈ k × k | f ( a , b ) = 0 } .
Example: Affine Plane Curve Let f ( x , y ) ∈ k [ x , y ] be an irreducible polynomial. The affine plane curve defined by f is X f = { ( a , b ) ∈ k × k | f ( a , b ) = 0 } . The function field of X f is k ( X f ) = k ( x )[ y ] / ( f ( x , y )) .
Example: Affine Plane Curve Let f ( x , y ) ∈ k [ x , y ] be an irreducible polynomial. The affine plane curve defined by f is X f = { ( a , b ) ∈ k × k | f ( a , b ) = 0 } . The function field of X f is k ( X f ) = k ( x )[ y ] / ( f ( x , y )) . We say that the curve X f corresponds to the ring k [ x , y ] / ( f ( x , y )).
Example: Affine Plane Curve Let f ( x , y ) ∈ k [ x , y ] be an irreducible polynomial. The affine plane curve defined by f is X f = { ( a , b ) ∈ k × k | f ( a , b ) = 0 } . The function field of X f is k ( X f ) = k ( x )[ y ] / ( f ( x , y )) . We say that the curve X f corresponds to the ring k [ x , y ] / ( f ( x , y )). For example, let f ( x , y ) = y − x 2 . If k = R , then visualize the curve as
Example: Affine Plane Curve f ( x , y ) = y − x 2 y x
Projective Plane Curve The projective plane over k , denoted P 2 ( k ), is defined to consist of points [ x 0 , x 1 , x 2 ], where x i ∈ k , and we declare two points to be equal if one is a nonzero scalar multiple of the other. A polynomial f ( x , y , z ) ∈ k [ x , y , z ] is homogeneous of degree d if f ( λ x , λ y , λ z ) = λ d f ( x , y , z ) for all λ ∈ k . Let f ( x , y , z ) ∈ k [ x , y , z ] be a homogeneous irreducible polynomial. The projective plane curve defined by f is X f = { [ x 0 , x 1 , x 2 ] ∈ P 2 ( k ) | f ( x 0 , x 1 , x 2 ) = 0 } .
Projective Plane Curve The projective plane over k , denoted P 2 ( k ), is defined to consist of points [ x 0 , x 1 , x 2 ], where x i ∈ k , and we declare two points to be equal if one is a nonzero scalar multiple of the other. A polynomial f ( x , y , z ) ∈ k [ x , y , z ] is homogeneous of degree d if f ( λ x , λ y , λ z ) = λ d f ( x , y , z ) for all λ ∈ k .
Projective Plane Curve The projective plane over k , denoted P 2 ( k ), is defined to consist of points [ x 0 , x 1 , x 2 ], where x i ∈ k , and we declare two points to be equal if one is a nonzero scalar multiple of the other. A polynomial f ( x , y , z ) ∈ k [ x , y , z ] is homogeneous of degree d if f ( λ x , λ y , λ z ) = λ d f ( x , y , z ) for all λ ∈ k . Let f ( x , y , z ) ∈ k [ x , y , z ] be a homogeneous irreducible polynomial. The projective plane curve defined by f is X f = { [ x 0 , x 1 , x 2 ] ∈ P 2 ( k ) | f ( x 0 , x 1 , x 2 ) = 0 } .
Projective Curve Generalize plane curves to an arbitrary curve
Projective Curve Generalize plane curves to an arbitrary curve Fact: Bijective correspondence between smooth projective curves X over k and function fields F over k
Projective Curve Generalize plane curves to an arbitrary curve Fact: Bijective correspondence between smooth projective curves X over k and function fields F over k Allow points at infinity
Projective Curve Generalize plane curves to an arbitrary curve Fact: Bijective correspondence between smooth projective curves X over k and function fields F over k Allow points at infinity Think of a plane curve
Group Actions Definition Let G be a group with identity e and let X be a set. A group action of G on X is a map G × X → X ( g , x ) �→ g . x satisfying ◮ e . x = x for all x ∈ X ◮ g . ( h . x ) = ( gh ) . x for all g , h ∈ G , for all x ∈ X
Example X , affine plane curve over R defined by f ( x , y ) = y − x 2 G = Z / 2 Z = { 0 , 1 } .
Example X , affine plane curve over R defined by f ( x , y ) = y − x 2 G = Z / 2 Z = { 0 , 1 } . ◮ By definition, 0 . ( a , b ) = ( a , b )
Example X , affine plane curve over R defined by f ( x , y ) = y − x 2 G = Z / 2 Z = { 0 , 1 } . ◮ By definition, 0 . ( a , b ) = ( a , b ) ◮ Define 1 . ( a , b ) = ( − a , b )
Example X , affine plane curve over R defined by f ( x , y ) = y − x 2 G = Z / 2 Z = { 0 , 1 } . ◮ By definition, 0 . ( a , b ) = ( a , b ) ◮ Define 1 . ( a , b ) = ( − a , b ) − →
Example X , affine plane curve over R defined by f ( x , y ) = y − x 2 G = Z / 2 Z = { 0 , 1 } . ◮ By definition, 0 . ( a , b ) = ( a , b ) ◮ Define 1 . ( a , b ) = ( − a , b ) − →
Module of Relative Differentials Let A be an algebra over a field k . Definition A derivation of A over k onto some space B is a map d : A → B so that 1. d ( x + y ) = d ( x ) + d ( y ) 2. d ( xy ) = xd ( y ) + d ( x ) y 3. d ( λ ) = 0 for x , y ∈ A , λ ∈ k .
Module of Relative Differentials Let A be an algebra over a field k . Definition A derivation of A over k onto some space B is a map d : A → B so that 1. d ( x + y ) = d ( x ) + d ( y ) 2. d ( xy ) = xd ( y ) + d ( x ) y 3. d ( λ ) = 0 for x , y ∈ A , λ ∈ k . Example A = k [ x ], define d : A → A by d ( f ( x )) = f ′ ( x )
Module of Relative Differentials Define the module of relative differentials of A over k to be an object Ω A / k together with a derivation of A over k , d : A → Ω A / k .
Module of Relative Differentials Define the module of relative differentials of A over k to be an object Ω A / k together with a derivation of A over k , d : A → Ω A / k . This space is generated by { d ( a ) | a ∈ A } . There could be relations!
Module of Relative Differentials Define the module of relative differentials of A over k to be an object Ω A / k together with a derivation of A over k , d : A → Ω A / k . This space is generated by { d ( a ) | a ∈ A } . There could be relations! Example A = k [ x ] Ω A / k = k [ x ] dx
Space of Holomorphic Differentials Idea: Make the module of relative differentials geometric
Space of Holomorphic Differentials Idea: Make the module of relative differentials geometric Let X be a smooth projective curve over a field k .
Space of Holomorphic Differentials Idea: Make the module of relative differentials geometric Let X be a smooth projective curve over a field k . We can cover X with two affine curves given by rings A 1 and A 2
Space of Holomorphic Differentials Idea: Make the module of relative differentials geometric Let X be a smooth projective curve over a field k . We can cover X with two affine curves given by rings A 1 and A 2 Define the space of holomorphic differentials to be H 0 ( X , Ω X ) = Ω A 1 / k ∩ Ω A 2 / k
Space of Holomorphic Differentials Idea: Make the module of relative differentials geometric Let X be a smooth projective curve over a field k . We can cover X with two affine curves given by rings A 1 and A 2 Define the space of holomorphic differentials to be H 0 ( X , Ω X ) = Ω A 1 / k ∩ Ω A 2 / k This space is a k -vector space. If a group G acts on X , then G acts on H 0 ( X , Ω X ) = ⇒ we get a representation of G
Space of Holomorphic Poly differentials We defined H 0 ( X , Ω X ) = Ω A 1 / k ∩ Ω A 2 / k
Space of Holomorphic Poly differentials We defined H 0 ( X , Ω X ) = Ω A 1 / k ∩ Ω A 2 / k Let m > 1. Define the space of holomorphic polydifferentials to be H 0 ( X , Ω ⊗ m X ) = Ω A 1 / k ⊗ A 1 · · · ⊗ A 1 Ω A 1 / k ∩ Ω A 2 / k ⊗ A 2 · · · ⊗ A 2 Ω A 2 / k . � �� � � �� � m times m times
Space of Holomorphic Poly differentials We defined H 0 ( X , Ω X ) = Ω A 1 / k ∩ Ω A 2 / k Let m > 1. Define the space of holomorphic polydifferentials to be H 0 ( X , Ω ⊗ m X ) = Ω A 1 / k ⊗ A 1 · · · ⊗ A 1 Ω A 1 / k ∩ Ω A 2 / k ⊗ A 2 · · · ⊗ A 2 Ω A 2 / k . � �� � � �� � m times m times As above, if a group G acts on X , we get a representation of G
Research on the Space of Holomorphic Polydifferentials Let k be a field, let X be a smooth projective curve over k , and let G be a group acting on X . Problem: Decompose H 0 ( X , Ω ⊗ m X ) into indecomposable representations
Research on the Space of Holomorphic Polydifferentials Let k be a field, let X be a smooth projective curve over k , and let G be a group acting on X . Problem: Decompose H 0 ( X , Ω ⊗ m X ) into indecomposable representations Variations: ◮ char( k ) = 0 or char( k ) = p ◮ Type of group G ◮ Ramification of the cover π : X → X / G ◮ Value of m
Previous Results ◮ char( k ) = 0, Chevalley and Weil, 1936 ◮ char( k ) = p , m = 1 ◮ Unramified cover, Tamagawa, 1951 ◮ Tamely ramified cover, Nakajima, 1986 ◮ Cyclic p -group, Valentini and Madan, 1981 ◮ Arbitrary p -group, Karanikolopoulos and Kontogeorgis, 2013 ◮ G has cyclic Sylow p -subgroups (includes the case when G = P ⋊ C ), m = 1, Bleher, Chinburg, and Kontogeorgis, 2017 ◮ char( k ) = p , m > 1, cyclic p -group, Karanikolopoulos, 2012
Example, computation of H 0 ( X , Ω ⊗ m X ) √ f ( t ) = t 9 − t F = k ( t )[ y ] / ( y 2 − f ( t )) ∼ = k ( t )( f ) X smooth projective curve over k with function field F , char( k ) = 3 Two important affine curves √ B 2 = k [ t − 1 ][ t − 5 √ B 1 = k [ t ][ f ] , f ] � � � � t − 2 k [ t − 1 ] + t 3 k [ t − 1 ] k [ t ] + k [ t ] Ω B 1 / k = √ dt Ω B 2 / k = √ dt f f H 0 ( X , Ω X ) = Ω B 1 / k ∩ Ω B 2 / k = ( k + kt + kt 2 + kt 3 ) dt √ f H 0 ( X , Ω ⊗ 2 X ) = (Ω B 1 / k ⊗ B 1 Ω B 1 / k ) ∩ (Ω B 2 / k ⊗ B 2 Ω B 2 / k ) � � ( k + kt ) 1 + ( k + kt + · · · + kt 6 )1 √ = ( dt ⊗ dt ) f f
Example, group action on F √ F = k ( t )( f ) G = Z / 3 Z ⋊ Z / 4 Z = � σ � ⋊ � ρ � Define action of G on F by √ √ σ. t = t + 1 σ. f = f √ f = ω − 1 √ ρ. t = − t ρ. f ω primitive fourth root of unity in k , extend multiplicatively to all of F
Example, “nice” bases � � ( k + kt ) 1 ⊕ ( k + kt + · · · + kt 6 )1 H 0 ( X , Ω ⊗ 2 √ X ) = ( dt ⊗ dt ) f f ρ. 1 1 1 = ω σ. ( dt ) = d ( σ. t ) = d ( t + 1) = dt √ = √ = ω − 1 √ √ ρ. f f f f = ω 2 ρ. 1 1 1 = − 1 ρ. ( dt ) = − dt ω − 2 √ √ √ f = 2 = √ f f f ρ. f t 5 t 2 − t + 1 σ − 1 σ − 1 − t 4 + t 3 + t 2 − t + 1 − t + 1 σ − 1 − 1 σ − 1 σ − 1 − t 3 + t t 6 + t 4 + t 2 σ − 1 − 1 0 σ − 1 σ − 1 σ − 1 0 0 0
Recall Consider G = Z / 3 Z ⋊ Z / 4 Z , char( k ) = 3 Let ω be a 4 th root of unity in k . We get 3 · 4 = 12 indecomposable representations visualized as follows: 1 1 0 � 1 � 1 (1) 0 1 1 0 1 0 0 1 ω 1 0 � ω � 1 ( ω ) 0 ω 1 0 ω 0 0 ω ω 2 1 0 � ω 2 � 1 ω 2 ( ω 2 ) 0 1 ω 2 0 ω 2 0 0 ω 3 1 0 � ω 3 � 1 ( ω 3 ) ω 3 0 1 ω 3 0 ω 3 0 0
Recall Consider G = Z / 3 Z ⋊ Z / 4 Z , char( k ) = 3 Let ω be a 4 th root of unity in k . We get 3 · 4 = 12 indecomposable representations visualized as follows: 1 1 0 � 1 � 1 (1) U 0 , 1 U 0 , 2 0 1 1 U 0 , 3 0 1 0 0 1 ω 1 0 � ω � 1 ( ω ) U 1 , 1 0 ω 1 U 1 , 2 U 1 , 3 0 ω 0 0 ω ω 2 1 0 � ω 2 � 1 ω 2 ( ω 2 ) U 2 , 1 U 2 , 2 0 1 U 2 , 3 ω 2 0 ω 2 0 0 ω 3 1 0 � ω 3 � 1 ( ω 3 ) ω 3 0 1 U 3 , 1 U 3 , 2 U 3 , 3 ω 3 0 ω 3 0 0
Example, “nice” bases � � ( k + kt ) 1 ⊕ ( k + kt + · · · + kt 6 )1 H 0 ( X , Ω ⊗ 2 √ X ) = ( dt ⊗ dt ) f f ρ. 1 1 1 = ω σ. ( dt ) = d ( σ. t ) = d ( t + 1) = dt √ = √ = ω − 1 √ √ ρ. f f f f = ω 2 ρ. 1 1 1 = − 1 ρ. ( dt ) = − dt ω − 2 √ √ √ f = 2 = √ f f f ρ. f t 5 t 2 − t + 1 σ − 1 σ − 1 − t 4 + t 3 + t 2 − t + 1 − t + 1 σ − 1 − 1 σ − 1 σ − 1 − t 3 + t t 6 + t 4 + t 2 σ − 1 − 1 0 σ − 1 σ − 1 σ − 1 0 0 0
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