Groups, Representation Theory, and Curves Adam Wood Department of - PowerPoint PPT Presentation

Groups, Representation Theory, and Curves Adam Wood Department of Mathematics University of Iowa MAA MathFest Great Talks for a General Audience August 3, 2019

Outline Introduction to (Modular) Representation Theory Representation Theory of Cyclic Groups Curves and Group Actions Space of Holomorphic Polydifferentials

Groups 12 11 1 10 2 9 3 8 4 7 5 6 8

Groups 12 11 1 10 2 9 3 8 4 7 5 6 8 + 1

Groups 12 11 1 10 2 9 3 8 4 7 5 6 8 + 2

Groups 12 11 1 10 2 9 3 8 4 7 5 6 8 + 3

Groups 12 11 1 10 2 9 3 8 4 7 5 6 8 + 4

Groups 12 11 1 10 2 9 3 8 4 7 5 6 8 + 5

Groups 12 11 1 10 2 9 3 8 4 7 5 6 8 + 6

Groups 12 11 1 10 2 9 3 8 4 7 5 6 8 + 6 = 2 mod 12

Groups 12 11 1 10 2 9 3 8 4 7 5 6 2 + 0 = 2

Groups 12 11 1 10 2 9 3 8 4 7 5 6 2

Groups 12 11 1 10 2 9 3 8 4 7 5 6 2 − 1

Groups 12 11 1 10 2 9 3 8 4 7 5 6 2 − 2

Groups 12 11 1 10 2 9 3 8 4 7 5 6 2 − 2 = 0

Groups Z = { . . . , − 2 , − 1 , 0 , 1 , 2 , . . . }

Groups Z = { . . . , − 2 , − 1 , 0 , 1 , 2 , . . . } 4 + 3 = 7

Groups Z = { . . . , − 2 , − 1 , 0 , 1 , 2 , . . . } 4 + 3 = 7 4 + 0 = 4

Groups Z = { . . . , − 2 , − 1 , 0 , 1 , 2 , . . . } 4 + 3 = 7 4 + 0 = 4 4 + ( − 4) = 0

Groups Z = { . . . , − 2 , − 1 , 0 , 1 , 2 , . . . } Z is a set with operation + 4 + 3 = 7 4 + 0 = 4 4 + ( − 4) = 0

Groups Z = { . . . , − 2 , − 1 , 0 , 1 , 2 , . . . } Z is a set with operation + 4 + 3 = 7 Closure 4 + 0 = 4 4 + ( − 4) = 0

Groups Z = { . . . , − 2 , − 1 , 0 , 1 , 2 , . . . } Z is a set with operation + 4 + 3 = 7 Closure 4 + 0 = 4 Identity 4 + ( − 4) = 0

Groups Z = { . . . , − 2 , − 1 , 0 , 1 , 2 , . . . } Z is a set with operation + 4 + 3 = 7 Closure 4 + 0 = 4 Identity 4 + ( − 4) = 0 Inverse

Definition of a Group Definition A group is a set G with an operation · so that ◮ g · h ∈ G for all g , h ∈ G (closure) ◮ ( a · b ) · c = a · ( b · c ) for all a , b , c ∈ G (associativity) ◮ There is an identity e ∈ G satisfying e · g = g for all g ∈ G (identity) ◮ For every g ∈ G , there is an inverse element, g − 1 , so that g · g − 1 = e = g − 1 · g (inverse)

Examples ◮ ( R − { 0 } , · )

Examples ◮ ( R − { 0 } , · ) ◮ ( Z , +)

Examples ◮ ( R − { 0 } , · ) ◮ ( Z , +) ◮ Z / 3 Z = { 0 , 1 , 2 } , addition modulo 3, 2 + 2 = 1

Examples ◮ ( R − { 0 } , · ) ◮ ( Z , +) ◮ Z / 3 Z = { 0 , 1 , 2 } , addition modulo 3, 2 + 2 = 1 ◮ ( Z , · ) is NOT a group

A Representation of a Group Definition Let G be a finite group and let k be a field. A representation of G is a group homomorphism ρ : G → GL ( V ) , where V is a vector space over k . For g ∈ G , we think of ρ ( g ) as an n × n matrix, where n is the dimension of V over k .

Example: Symmetric Group Let G = S 3 . The permutation matrix ( a ij ) associated to σ ∈ S 3 satisfies � 1 if σ ( i ) = j a ij = 0 otherwise .

Example: Symmetric Group Let G = S 3 . The permutation matrix ( a ij ) associated to σ ∈ S 3 satisfies � 1 if σ ( i ) = j a ij = 0 otherwise . For example, the permutation matrix associated to (1 2) is   0 1 0  . 1 0 0  0 0 1

Example: Symmetric Group Let G = S 3 . The permutation matrix ( a ij ) associated to σ ∈ S 3 satisfies � 1 if σ ( i ) = j a ij = 0 otherwise . For example, the permutation matrix associated to (1 2) is   0 1 0  . 1 0 0  0 0 1 Let M σ denote the permutation matrix associated to σ . Then, ρ ( σ ) = M σ defines a representation of G called the permutation representation of G .

Example: Dihedral Group Consider G = D 8 = � r , s | r 4 = 1 = s 2 , srs = r − 1 � .

Example: Dihedral Group Consider G = D 8 = � r , s | r 4 = 1 = s 2 , srs = r − 1 � . � 0 � rotation by − 1 r 90 ◦ 1 0 � − 1 � reflection 0 s about y -axis 0 1

Example: Dihedral Group Consider G = D 8 = � r , s | r 4 = 1 = s 2 , srs = r − 1 � . � 0 � rotation by − 1 r 90 ◦ 1 0 � − 1 � reflection 0 s about y -axis 0 1

Example: Dihedral Group Consider G = D 8 = � r , s | r 4 = 1 = s 2 , srs = r − 1 � . � 0 � rotation by − 1 r 90 ◦ 1 0 � − 1 � reflection 0 s about y -axis 0 1

Example: Dihedral Group Consider G = D 8 = � r , s | r 4 = 1 = s 2 , srs = r − 1 � . � 0 � rotation by − 1 r 90 ◦ 1 0 � − 1 � reflection 0 s about y -axis 0 1 � 0 � � − 1 � − 1 0 ρ : G → M 2 ( C ) defined by ρ ( r ) = and ρ ( s ) = 1 0 0 1 defines a representation of G .

Subrepresentations and Direct Sums Definition (Direct Sum) � V � 0 V ⊕ W = 0 W

Subrepresentations and Direct Sums Definition (Direct Sum) � V � 0 V ⊕ W = 0 W V and W are subrepresentations of the direct sum of V and W

Subrepresentations and Direct Sums Definition (Direct Sum) � V � 0 V ⊕ W = 0 W V and W are subrepresentations of the direct sum of V and W BUT, there are more subrepresentations

Irreducible and Indecomposable Representations Definition A representation V of G is called simple or irreducible if the only subrepresentations of V are 0 and V .

Irreducible and Indecomposable Representations Definition A representation V of G is called simple or irreducible if the only subrepresentations of V are 0 and V . Definition A representation V is called indecomposable if whenever V = U ⊕ W is a direct sum decomposition of V , either U or W is zero.

Irreducible and Indecomposable Representations Definition A representation V of G is called simple or irreducible if the only subrepresentations of V are 0 and V . Definition A representation V is called indecomposable if whenever V = U ⊕ W is a direct sum decomposition of V , either U or W is zero. Note that simple = ⇒ indecomposable but

Irreducible and Indecomposable Representations Definition A representation V of G is called simple or irreducible if the only subrepresentations of V are 0 and V . Definition A representation V is called indecomposable if whenever V = U ⊕ W is a direct sum decomposition of V , either U or W is zero. Note that simple = ⇒ indecomposable but indecomposable � = ⇒ simple .

Characteristic of a Field Z / p Z = { 0 , 1 , . . . , p − 1 } Two types of fields: Characteristic zero Prime characteristic Think of: C , Q F p

Characteristic of a Field F p = { 0 , 1 , . . . , p − 1 }

Characteristic of a Field F p = { 0 , 1 , . . . , p − 1 } Two types of fields: ◦ Characteristic zero ◦ Prime characteristic

Characteristic of a Field F p = { 0 , 1 , . . . , p − 1 } Two types of fields: ◦ Characteristic zero ◦ Prime characteristic Think of: ◦ C , Q ◦ F p

Modular Representation Theory Study of representations of G over k , field of prime characteristic

Modular Representation Theory Study of representations of G over k , field of prime characteristic Every representation can be written as a direct sum of indecomposable representations.

Modular Representation Theory Study of representations of G over k , field of prime characteristic 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 ◮ Can be extended to other groups

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 Trivial Dimension 2 Dimension 3 Representation Representation Representation

Visualization of Representations of the Cyclic Group Let G be a cyclic group of order p n and let k be a field of characteristic p . Brauer Tree: ◦ • p n − 1 Quiver: • α α p n = 0

Curves and Group Actions Goal: Define a representation of a group using geometry

Curves and Group Actions Goal: Define a representation of a group using geometry ◮ Define an algebraic curve

Curves and Group Actions Goal: Define a representation of a group using geometry ◮ Define an algebraic curve ◮ Group actions on curves

Curves and Group Actions Goal: Define a representation of a group using geometry ◮ Define an algebraic curve ◮ Group actions on curves ◮ Define a representation using geometry