Character Theory of Finite Groups

Elias Sink and Allen Wang

Mentor: Chris Ryba

PRIMES Conference: May 18th, 2019

Elias Sink and Allen Wang Character Theory of Finite Groups PRIMES Conference 1 / 13 Representation theory gives us a nice way of translating abstract relations into an easier language.

We will focus on the finite representation of groups and work with vector spaces over C. We pick C because it is algebraically closed and has characteristic 0.

Motivation

The only math that we truly understand is linear algebra.

Elias Sink and Allen Wang Character Theory of Finite Groups PRIMES Conference 2 / 13 We will focus on the finite representation of groups and work with vector spaces over C. We pick C because it is algebraically closed and has characteristic 0.

Motivation

The only math that we truly understand is linear algebra.

Representation theory gives us a nice way of translating abstract relations into an easier language.

Elias Sink and Allen Wang Character Theory of Finite Groups PRIMES Conference 2 / 13 Motivation

The only math that we truly understand is linear algebra.

Representation theory gives us a nice way of translating abstract relations into an easier language.

We will focus on the finite representation of groups and work with vector spaces over C. We pick C because it is algebraically closed and has characteristic 0.

Elias Sink and Allen Wang Character Theory of Finite Groups PRIMES Conference 2 / 13 For example, the regular representation of a group G is the representation (C[G], ρ) where C[G] is the vector space freely generated by G and ρ(g) is multiplication by g on the left. Definition

Given a group G and representations V and W , let HomG (V , W ) be the linear maps φ : V → W with φρV (g) = ρW (g)φ.

Basic Definitions

Definition A representation of a group G is the pair (V , ρ) where V is a vector space and ρ is a group homomorphism from G → GL(V ), i.e. ρ(g1)ρ(g2) = ρ(g1g2).

Elias Sink and Allen Wang Character Theory of Finite Groups PRIMES Conference 3 / 13 Definition

Given a group G and representations V and W , let HomG (V , W ) be the linear maps φ : V → W with φρV (g) = ρW (g)φ.

Basic Definitions

Definition A representation of a group G is the pair (V , ρ) where V is a vector space and ρ is a group homomorphism from G → GL(V ), i.e. ρ(g1)ρ(g2) = ρ(g1g2).

For example, the regular representation of a group G is the representation (C[G], ρ) where C[G] is the vector space freely generated by G and ρ(g) is multiplication by g on the left.

Elias Sink and Allen Wang Character Theory of Finite Groups PRIMES Conference 3 / 13 Basic Definitions

Definition A representation of a group G is the pair (V , ρ) where V is a vector space and ρ is a group homomorphism from G → GL(V ), i.e. ρ(g1)ρ(g2) = ρ(g1g2).

For example, the regular representation of a group G is the representation (C[G], ρ) where C[G] is the vector space freely generated by G and ρ(g) is multiplication by g on the left. Definition

Given a group G and representations V and W , let HomG (V , W ) be the linear maps φ : V → W with φρV (g) = ρW (g)φ.

Elias Sink and Allen Wang Character Theory of Finite Groups PRIMES Conference 3 / 13 Lemma (Schur) Let V and W be simple representations of G. If they are distinct, then ∼ dim HomG (V , W ) = 0. If V = W , then dim HomG (V , W ) = 1.

Let nonzero φ : V → W be in HomG (V , W ). If V is simple, then φ is injective. If W is simple, then φ is surjective. Theorem (Maschke) Let V be any representation of G. Then V is the direct sum of simple representations of G.

Basic Results

Definition A representation V of G is simple (or irreducible) if there is no proper nonzero subrepresentation of V .

Elias Sink and Allen Wang Character Theory of Finite Groups PRIMES Conference 4 / 13 Let nonzero φ : V → W be in HomG (V , W ). If V is simple, then φ is injective. If W is simple, then φ is surjective. Theorem (Maschke) Let V be any representation of G. Then V is the direct sum of simple representations of G.

Basic Results

Definition A representation V of G is simple (or irreducible) if there is no proper nonzero subrepresentation of V .

Lemma (Schur) Let V and W be simple representations of G. If they are distinct, then ∼ dim HomG (V , W ) = 0. If V = W , then dim HomG (V , W ) = 1.

Elias Sink and Allen Wang Character Theory of Finite Groups PRIMES Conference 4 / 13 Theorem (Maschke) Let V be any representation of G. Then V is the direct sum of simple representations of G.

Basic Results

Definition A representation V of G is simple (or irreducible) if there is no proper nonzero subrepresentation of V .

Lemma (Schur) Let V and W be simple representations of G. If they are distinct, then ∼ dim HomG (V , W ) = 0. If V = W , then dim HomG (V , W ) = 1.

Let nonzero φ : V → W be in HomG (V , W ). If V is simple, then φ is injective. If W is simple, then φ is surjective.

Elias Sink and Allen Wang Character Theory of Finite Groups PRIMES Conference 4 / 13 Basic Results

Definition A representation V of G is simple (or irreducible) if there is no proper nonzero subrepresentation of V .

Lemma (Schur) Let V and W be simple representations of G. If they are distinct, then ∼ dim HomG (V , W ) = 0. If V = W , then dim HomG (V , W ) = 1.

Let nonzero φ : V → W be in HomG (V , W ). If V is simple, then φ is injective. If W is simple, then φ is surjective. Theorem (Maschke) Let V be any representation of G. Then V is the direct sum of simple representations of G.

Elias Sink and Allen Wang Character Theory of Finite Groups PRIMES Conference 4 / 13 Example (S3)

We give examples of irreducible representations of S3. The trivial representation, C+, which sends all g to 1. The sign representation, C−, which sends all elements to sgn(g) ∈ {−1, +1}. 2 The space (a, b, c): a + b + c = 0, C , where g acts by permutation of coordinates.

Examples of Representations

Example (C3) 3 The regular representation of C3 is C where the action of g ∈ C3 is cyclically permuting the coordinates. The space (a, a, a) is the trivial representation. The space (a, b, c): a + b + c = 0 is a two-dimensional subrepresentation.

Elias Sink and Allen Wang Character Theory of Finite Groups PRIMES Conference 5 / 13 Examples of Representations

Example (C3) 3 The regular representation of C3 is C where the action of g ∈ C3 is cyclically permuting the coordinates. The space (a, a, a) is the trivial representation. The space (a, b, c): a + b + c = 0 is a two-dimensional subrepresentation.

Example (S3)

We give examples of irreducible representations of S3. The trivial representation, C+, which sends all g to 1. The sign representation, C−, which sends all elements to sgn(g) ∈ {−1, +1}. 2 The space (a, b, c): a + b + c = 0, C , where g acts by permutation of coordinates.

Elias Sink and Allen Wang Character Theory of Finite Groups PRIMES Conference 5 / 13 Note that since Tr(MgM−1) = Tr(g), so character is independent of the −1 choice of basis for V . Similarly, Tr(g2g1g2 ) = Tr(g1), so χ is constant on conjugacy classes, i.e. it is a class function. Lemma

If V and W are representations of G, then χV ⊕W = χV + χW .

  ρV (g) 0 χV ⊕W (g) = Tr = Tr(ρV (g)) + Tr(ρW (g)) 0 ρW (g)

= χV (g) + χW (g)

Definition of Characters

Definition The character χV of a representation V is the function χ : G → C defined by χ(g) = Tr(ρ(g)).

Elias Sink and Allen Wang Character Theory of Finite Groups PRIMES Conference 6 / 13 Lemma

If V and W are representations of G, then χV ⊕W = χV + χW .

  ρV (g) 0 χV ⊕W (g) = Tr = Tr(ρV (g)) + Tr(ρW (g)) 0 ρW (g)

= χV (g) + χW (g)

Definition of Characters

Definition The character χV of a representation V is the function χ : G → C defined by χ(g) = Tr(ρ(g)).

Note that since Tr(MgM−1) = Tr(g), so character is independent of the −1 choice of basis for V . Similarly, Tr(g2g1g2 ) = Tr(g1), so χ is constant on conjugacy classes, i.e. it is a class function.

Elias Sink and Allen Wang Character Theory of Finite Groups PRIMES Conference 6 / 13   ρV (g) 0 χV ⊕W (g) = Tr = Tr(ρV (g)) + Tr(ρW (g)) 0 ρW (g)

= χV (g) + χW (g)

Definition of Characters

Definition The character χV of a representation V is the function χ : G → C defined by χ(g) = Tr(ρ(g)).

Note that since Tr(MgM−1) = Tr(g), so character is independent of the −1 choice of basis for V . Similarly, Tr(g2g1g2 ) = Tr(g1), so χ is constant on conjugacy classes, i.e. it is a class function. Lemma

If V and W are representations of G, then χV ⊕W = χV + χW .

Elias Sink and Allen Wang Character Theory of Finite Groups PRIMES Conference 6 / 13 Definition of Characters

Definition The character χV of a representation V is the function χ : G → C defined by χ(g) = Tr(ρ(g)).

Note that since Tr(MgM−1) = Tr(g), so character is independent of the −1 choice of basis for V . Similarly, Tr(g2g1g2 ) = Tr(g1), so χ is constant on conjugacy classes, i.e. it is a class function. Lemma

If V and W are representations of G, then χV ⊕W = χV + χW .

  ρV (g) 0 χV ⊕W (g) = Tr = Tr(ρV (g)) + Tr(ρW (g)) 0 ρW (g)

= χV (g) + χW (g)

Elias Sink and Allen Wang Character Theory of Finite Groups PRIMES Conference 6 / 13 The sign representation, C−, has character χ(g) = sgn(g). The space (a, b, c): a + b + c = 0 where g acts by permutation of 2 coordinates is the mean zero representation, C . Thus, χ(g) is one less than the number of fixed points of g.

Examples of Characters

Example (S3)

The trivial representation, C+, has character χ(g) = 1.

Elias Sink and Allen Wang Character Theory of Finite Groups PRIMES Conference 7 / 13 The space (a, b, c): a + b + c = 0 where g acts by permutation of 2 coordinates is the mean zero representation, C . Thus, χ(g) is one less than the number of fixed points of g.

Examples of Characters

Example (S3)

The trivial representation, C+, has character χ(g) = 1. The sign representation, C−, has character χ(g) = sgn(g).

Elias Sink and Allen Wang Character Theory of Finite Groups PRIMES Conference 7 / 13 Examples of Characters

Example (S3)

The trivial representation, C+, has character χ(g) = 1. The sign representation, C−, has character χ(g) = sgn(g). The space (a, b, c): a + b + c = 0 where g acts by permutation of 2 coordinates is the mean zero representation, C . Thus, χ(g) is one less than the number of fixed points of g.

Elias Sink and Allen Wang Character Theory of Finite Groups PRIMES Conference 7 / 13 It can be shown from Maschke’s Theorem that characters of simple representations are linearly independent and span the vector space Fc (G, C) of class functions G → C. Define an inner product (−, −) on Fc (G, C) by 1 X (f , f ) = f (g)f (g) 1 2 |G| 1 2 g∈G

or, letting {Ci } be the conjugacy classes of G,

X |Ci | f (C )f (C ). |G| 1 i 2 i i

Inner Product

What kind of structure do characters have?

Elias Sink and Allen Wang Character Theory of Finite Groups PRIMES Conference 8 / 13 Define an inner product (−, −) on Fc (G, C) by 1 X (f , f ) = f (g)f (g) 1 2 |G| 1 2 g∈G

or, letting {Ci } be the conjugacy classes of G,

X |Ci | f (C )f (C ). |G| 1 i 2 i i

Inner Product

What kind of structure do characters have? It can be shown from Maschke’s Theorem that characters of simple representations are linearly independent and span the vector space Fc (G, C) of class functions G → C.

Elias Sink and Allen Wang Character Theory of Finite Groups PRIMES Conference 8 / 13 Inner Product

What kind of structure do characters have? It can be shown from Maschke’s Theorem that characters of simple representations are linearly independent and span the vector space Fc (G, C) of class functions G → C. Define an inner product (−, −) on Fc (G, C) by 1 X (f , f ) = f (g)f (g) 1 2 |G| 1 2 g∈G

or, letting {Ci } be the conjugacy classes of G,

X |Ci | f (C )f (C ). |G| 1 i 2 i i

Elias Sink and Allen Wang Character Theory of Finite Groups PRIMES Conference 8 / 13 Proof sketch: it can be shown that 1 X χ (g)χ (g) = dim Hom (W , V ). |G| V W G g∈G

The result then follows immediately from Schur’s lemma. Thus this basis is orthonormal with respect to (−, −).

Orthogonality Relations

Theorem (Orthogonality by rows) ( 1 V =∼ W For V , W simple, (χV , χW ) = 0 V =6∼ W

Elias Sink and Allen Wang Character Theory of Finite Groups PRIMES Conference 9 / 13 Thus this basis is orthonormal with respect to (−, −).

Orthogonality Relations

Theorem (Orthogonality by rows) ( 1 V =∼ W For V , W simple, (χV , χW ) = 0 V =6∼ W

Proof sketch: it can be shown that 1 X χ (g)χ (g) = dim Hom (W , V ). |G| V W G g∈G

The result then follows immediately from Schur’s lemma.

Elias Sink and Allen Wang Character Theory of Finite Groups PRIMES Conference 9 / 13 Orthogonality Relations

Theorem (Orthogonality by rows) ( 1 V =∼ W For V , W simple, (χV , χW ) = 0 V =6∼ W

Proof sketch: it can be shown that 1 X χ (g)χ (g) = dim Hom (W , V ). |G| V W G g∈G

The result then follows immediately from Schur’s lemma. Thus this basis is orthonormal with respect to (−, −).

Elias Sink and Allen Wang Character Theory of Finite Groups PRIMES Conference 9 / 13 P Some calculation gives (δi , δj ) = V χV (Ci )χV (Cj ), where the sum is over simple representations.This leads to Theorem (Orthogonality by columns) ( P |G|/|Ci | i = j χV (Ci )χV (Cj ) = V 0 i 6= j

Orthogonality Relations, Cont.

p A different orthonormal basis is given by { |G|/|Ci |δi }, where ( 1 g ∈ Ci δi (g) = 0 g 6∈ Ci

Elias Sink and Allen Wang Character Theory of Finite Groups PRIMES Conference 10 / 13 This leads to Theorem (Orthogonality by columns) ( P |G|/|Ci | i = j χV (Ci )χV (Cj ) = V 0 i 6= j

Orthogonality Relations, Cont.

p A different orthonormal basis is given by { |G|/|Ci |δi }, where ( 1 g ∈ Ci δi (g) = 0 g 6∈ Ci P Some calculation gives (δi , δj ) = V χV (Ci )χV (Cj ), where the sum is over simple representations.

Elias Sink and Allen Wang Character Theory of Finite Groups PRIMES Conference 10 / 13 Orthogonality Relations, Cont.

p A different orthonormal basis is given by { |G|/|Ci |δi }, where ( 1 g ∈ Ci δi (g) = 0 g 6∈ Ci P Some calculation gives (δi , δj ) = V χV (Ci )χV (Cj ), where the sum is over simple representations.This leads to Theorem (Orthogonality by columns) ( P |G|/|Ci | i = j χV (Ci )χV (Cj ) = V 0 i 6= j

Elias Sink and Allen Wang Character Theory of Finite Groups PRIMES Conference 10 / 13 Character Tables

These data can be summarized in a character table. Rows are indexed by simples, columns by conjugacy classes. The number in row V and column C is χV (C). A row giving the size of each conjugacy class is also included.

Example (S3) 3 1 1 1 S3 1 1 2 3 # 1 3 2 C+ 1 1 1 C− 1 −1 1 2 C 2 0 −1

Elias Sink and Allen Wang Character Theory of Finite Groups PRIMES Conference 11 / 13 Furthermore, this is accomplished with a easy, concrete computation. Operations such as taking quotients and tensor products are similarly tractable with this machine. Thus the character table of a finite group gives an essentially complete description of its representation theory as well as a powerful computational tool for working with ostensibly abstract objects.

Conclusion

This information, together with Schur’s lemma and Maschke’s theorem, can be used to extract the simple summands (with multiplicity) of any representation of G, which determine it up to isomorphism.

Elias Sink and Allen Wang Character Theory of Finite Groups PRIMES Conference 12 / 13 Thus the character table of a finite group gives an essentially complete description of its representation theory as well as a powerful computational tool for working with ostensibly abstract objects.

Conclusion

This information, together with Schur’s lemma and Maschke’s theorem, can be used to extract the simple summands (with multiplicity) of any representation of G, which determine it up to isomorphism. Furthermore, this is accomplished with a easy, concrete computation. Operations such as taking quotients and tensor products are similarly tractable with this machine.

Elias Sink and Allen Wang Character Theory of Finite Groups PRIMES Conference 12 / 13 Conclusion

This information, together with Schur’s lemma and Maschke’s theorem, can be used to extract the simple summands (with multiplicity) of any representation of G, which determine it up to isomorphism. Furthermore, this is accomplished with a easy, concrete computation. Operations such as taking quotients and tensor products are similarly tractable with this machine. Thus the character table of a finite group gives an essentially complete description of its representation theory as well as a powerful computational tool for working with ostensibly abstract objects.

Elias Sink and Allen Wang Character Theory of Finite Groups PRIMES Conference 12 / 13 Our mentor, Chris Ryba Dr. Gerovitch and the MIT PRIMES program Prof. Pavel Etingof for his course notes and book Our parents

Acknowledgements

We would like to thank:

Elias Sink and Allen Wang Character Theory of Finite Groups PRIMES Conference 13 / 13 Dr. Gerovitch and the MIT PRIMES program Prof. Pavel Etingof for his course notes and book Our parents

Acknowledgements

We would like to thank: Our mentor, Chris Ryba

Elias Sink and Allen Wang Character Theory of Finite Groups PRIMES Conference 13 / 13 Prof. Pavel Etingof for his course notes and book Our parents

Acknowledgements

We would like to thank: Our mentor, Chris Ryba Dr. Gerovitch and the MIT PRIMES program

Elias Sink and Allen Wang Character Theory of Finite Groups PRIMES Conference 13 / 13 Our parents

Acknowledgements

We would like to thank: Our mentor, Chris Ryba Dr. Gerovitch and the MIT PRIMES program Prof. Pavel Etingof for his course notes and book

Elias Sink and Allen Wang Character Theory of Finite Groups PRIMES Conference 13 / 13 Acknowledgements

We would like to thank: Our mentor, Chris Ryba Dr. Gerovitch and the MIT PRIMES program Prof. Pavel Etingof for his course notes and book Our parents

Elias Sink and Allen Wang Character Theory of Finite Groups PRIMES Conference 13 / 13