Certain Monomial Characters and Their Subnormal Constituents - - PowerPoint PPT Presentation

Certain Monomial Characters and Their Subnormal Constituents

Carolina Vallejo

Universitat de Val` encia

• St. Andrews, August 2013

Carolina Vallejo (Universitat de Val` encia) Certain Monomial Characters

• St. Andrews, August 2013

1 / 15

This is a joint work with G. Navarro.

Carolina Vallejo (Universitat de Val` encia) Certain Monomial Characters

• St. Andrews, August 2013

2 / 15

Introduction

Introduction

Carolina Vallejo (Universitat de Val` encia) Certain Monomial Characters

• St. Andrews, August 2013

3 / 15

Introduction

Let ● be a group. A character ✤ ✷ ■rr✭●✮ is said to be monomial if there exist a subgroup ❯ ✒ ● and a linear ✕ ✷ ■rr✭❯✮, such that ✤ ❂ ✕●✿

• Carolina Vallejo (Universitat de Val`

encia) Certain Monomial Characters

• St. Andrews, August 2013

4 / 15

Introduction

Let ● be a group. A character ✤ ✷ ■rr✭●✮ is said to be monomial if there exist a subgroup ❯ ✒ ● and a linear ✕ ✷ ■rr✭❯✮, such that ✤ ❂ ✕●✿ A group ● is said to be monomial if all its irreducible characters are monomial.

Carolina Vallejo (Universitat de Val` encia) Certain Monomial Characters

• St. Andrews, August 2013

4 / 15

Introduction

There are few results guaranteeing that a given character of a group is monomial.

• Carolina Vallejo (Universitat de Val`

encia) Certain Monomial Characters

• St. Andrews, August 2013

5 / 15

Introduction

There are few results guaranteeing that a given character of a group is monomial.

Theorem

Let ● be a supersolvable group. Then all irreducible characters of ● are monomial.

Carolina Vallejo (Universitat de Val` encia) Certain Monomial Characters

• St. Andrews, August 2013

5 / 15

Introduction

There are few results guaranteeing that a given character of a group is monomial.

Theorem

Let ● be a supersolvable group. Then all irreducible characters of ● are monomial. Thus, supersolvable groups are monomial groups.

Carolina Vallejo (Universitat de Val` encia) Certain Monomial Characters

• St. Andrews, August 2013

5 / 15

Introduction

There are few results guaranteeing that a given character of a group is monomial.

Theorem

Let ● be a supersolvable group. Then all irreducible characters of ● are monomial. Thus, supersolvable groups are monomial groups. But this result depends more on the structure of the group than on characters themselves.

Carolina Vallejo (Universitat de Val` encia) Certain Monomial Characters

• St. Andrews, August 2013

5 / 15

Introduction

An interesting result.

• ✤ ✷ ■rr✭●✮

✤

Carolina Vallejo (Universitat de Val` encia) Certain Monomial Characters

• St. Andrews, August 2013

6 / 15

Introduction

An interesting result.

Theorem (Gow)

Let ● be a solvable group. Suppose that ✤ ✷ ■rr✭●✮ takes real values and has odd degree. Then ✤ is rational-valued and monomial.

Carolina Vallejo (Universitat de Val` encia) Certain Monomial Characters

• St. Andrews, August 2013

6 / 15

Introduction

An interesting result.

Theorem (Gow)

Let ● be a solvable group. Suppose that ✤ ✷ ■rr✭●✮ takes real values and has odd degree. Then ✤ is rational-valued and monomial. We give a monomiality criterium which also deals with fields of values and degrees of characters.

Carolina Vallejo (Universitat de Val` encia) Certain Monomial Characters

• St. Andrews, August 2013

6 / 15

Introduction

Notation: For ♥ an integer, we write ◗♥ ❂ ◗✭✘✮❀ where ✘ is a primitive ♥th root of unity.

• ♣

❥

• ✭P✮ ✿ P❥

P ✷ ❙②❧♣✭●✮ ♣ ✤ ✷ ■rr✭●✮ ♣ ✤ ◗❥●❥♣ ✤ ♣ ❂ ✷

Carolina Vallejo (Universitat de Val` encia) Certain Monomial Characters

• St. Andrews, August 2013

7 / 15

Introduction

Notation: For ♥ an integer, we write ◗♥ ❂ ◗✭✘✮❀ where ✘ is a primitive ♥th root of unity.

Theorem A

Let ● be a ♣-solvable group. Assume that ❥N●✭P✮ ✿ P❥ is odd, where P ✷ ❙②❧♣✭●✮ for some prime ♣. If ✤ ✷ ■rr✭●✮ has degree not divisible by ♣ and the values of ✤ are contained in the cyclotomic extension ◗❥●❥♣, then ✤ is monomial. ♣ ❂ ✷

Carolina Vallejo (Universitat de Val` encia) Certain Monomial Characters

• St. Andrews, August 2013

7 / 15

Introduction

Notation: For ♥ an integer, we write ◗♥ ❂ ◗✭✘✮❀ where ✘ is a primitive ♥th root of unity.

Theorem A

Let ● be a ♣-solvable group. Assume that ❥N●✭P✮ ✿ P❥ is odd, where P ✷ ❙②❧♣✭●✮ for some prime ♣. If ✤ ✷ ■rr✭●✮ has degree not divisible by ♣ and the values of ✤ are contained in the cyclotomic extension ◗❥●❥♣, then ✤ is monomial. When ♣ ❂ ✷, we can recover Gow’s result from Theorem A.

Carolina Vallejo (Universitat de Val` encia) Certain Monomial Characters

• St. Andrews, August 2013

7 / 15

Introduction

The hypothesis about the index ❥N●✭P✮ ✿ P❥ is necessary. ❆✻

Carolina Vallejo (Universitat de Val` encia) Certain Monomial Characters

• St. Andrews, August 2013

8 / 15

Introduction

The hypothesis about the index ❥N●✭P✮ ✿ P❥ is necessary. For instance, the group SL(2,3) and the prime p=3. ❆✻

Carolina Vallejo (Universitat de Val` encia) Certain Monomial Characters

• St. Andrews, August 2013

8 / 15

Introduction

The hypothesis about the index ❥N●✭P✮ ✿ P❥ is necessary. For instance, the group SL(2,3) and the prime p=3. The solvability hypothesis is necessary in both Gow’s and Theorem A. ❆✻

Carolina Vallejo (Universitat de Val` encia) Certain Monomial Characters

• St. Andrews, August 2013

8 / 15

Introduction

The hypothesis about the index ❥N●✭P✮ ✿ P❥ is necessary. For instance, the group SL(2,3) and the prime p=3. The solvability hypothesis is necessary in both Gow’s and Theorem A. The alternating group ❆✻ is a counterexample in both cases.

Carolina Vallejo (Universitat de Val` encia) Certain Monomial Characters

• St. Andrews, August 2013

8 / 15

❇✙ Theory

❇✙ Theory

Carolina Vallejo (Universitat de Val` encia) Certain Monomial Characters

• St. Andrews, August 2013

9 / 15

❇✙ Theory

We say that ✤ ✷ ■rr✭●✮ is a ✙-special character of ●, if ✭❛✮ ✤✭✶✮ ✙ ✭❜✮ ◆ ✴ ✴ ● ✤◆ ✙

✙

• ✙
• Carolina Vallejo (Universitat de Val`

encia) Certain Monomial Characters

• St. Andrews, August 2013

10 / 15

❇✙ Theory

We say that ✤ ✷ ■rr✭●✮ is a ✙-special character of ●, if ✭❛✮ ✤✭✶✮ is a ✙-number. ✭❜✮ ◆ ✴ ✴ ● ✤◆ ✙

✙

• ✙
• Carolina Vallejo (Universitat de Val`

encia) Certain Monomial Characters

• St. Andrews, August 2013

10 / 15

❇✙ Theory

We say that ✤ ✷ ■rr✭●✮ is a ✙-special character of ●, if ✭❛✮ ✤✭✶✮ is a ✙-number. ✭❜✮ For every subnormal subgroup ◆ ✴ ✴ ●, the order of all the irreducible constituents of ✤◆ is a ✙-number.

✙

• ✙
• Carolina Vallejo (Universitat de Val`

encia) Certain Monomial Characters

• St. Andrews, August 2013

10 / 15

❇✙ Theory

We say that ✤ ✷ ■rr✭●✮ is a ✙-special character of ●, if ✭❛✮ ✤✭✶✮ is a ✙-number. ✭❜✮ For every subnormal subgroup ◆ ✴ ✴ ●, the order of all the irreducible constituents of ✤◆ is a ✙-number. A B✙ character of a group ●

• ✙
• Carolina Vallejo (Universitat de Val`

encia) Certain Monomial Characters

• St. Andrews, August 2013

10 / 15

❇✙ Theory

We say that ✤ ✷ ■rr✭●✮ is a ✙-special character of ●, if ✭❛✮ ✤✭✶✮ is a ✙-number. ✭❜✮ For every subnormal subgroup ◆ ✴ ✴ ●, the order of all the irreducible constituents of ✤◆ is a ✙-number. A B✙ character of a group ● may be thought as an irreducible character of ● induced from a ✙-special character of some subgroup

• f ●.

Carolina Vallejo (Universitat de Val` encia) Certain Monomial Characters

• St. Andrews, August 2013

10 / 15

❇✙ Theory

We say that ✤ ✷ ■rr✭●✮ is a ✙-special character of ●, if ✭❛✮ ✤✭✶✮ is a ✙-number. ✭❜✮ For every subnormal subgroup ◆ ✴ ✴ ●, the order of all the irreducible constituents of ✤◆ is a ✙-number. A B✙ character of a group ● may be thought as an irreducible character of ● induced from a ✙-special character of some subgroup

• f ●. (True in groups of odd order).

Carolina Vallejo (Universitat de Val` encia) Certain Monomial Characters

• St. Andrews, August 2013

10 / 15

Main results

Main results

Carolina Vallejo (Universitat de Val` encia) Certain Monomial Characters

• St. Andrews, August 2013

11 / 15

Main results

Theorem B

Let ● be a ♣-solvable group. Assume that ❥N●✭P✮ ✿ P❥ is odd, where P ✷ ❙②❧♣✭●✮ for some prime ♣. If ✤ ✷ ■rr✭●✮ has degree not divisible by ♣ and its values are contained in the cyclotomic extension ◗❥●❥♣ , then ✤ is a ❇♣ character of ●. ❇♣ ♣

Carolina Vallejo (Universitat de Val` encia) Certain Monomial Characters

• St. Andrews, August 2013

12 / 15

Main results

Theorem B

Let ● be a ♣-solvable group. Assume that ❥N●✭P✮ ✿ P❥ is odd, where P ✷ ❙②❧♣✭●✮ for some prime ♣. If ✤ ✷ ■rr✭●✮ has degree not divisible by ♣ and its values are contained in the cyclotomic extension ◗❥●❥♣ , then ✤ is a ❇♣ character of ●. Notice that ❇♣ characters with degree not divisible by ♣ are monomial.

Carolina Vallejo (Universitat de Val` encia) Certain Monomial Characters

• St. Andrews, August 2013

12 / 15

Main results

Theorem B

Let ● be a ♣-solvable group. Assume that ❥N●✭P✮ ✿ P❥ is odd, where P ✷ ❙②❧♣✭●✮ for some prime ♣. If ✤ ✷ ■rr✭●✮ has degree not divisible by ♣ and its values are contained in the cyclotomic extension ◗❥●❥♣ , then ✤ is a ❇♣ character of ●. Notice that ❇♣ characters with degree not divisible by ♣ are monomial.Thus Theorem B implies Theorem A.

Carolina Vallejo (Universitat de Val` encia) Certain Monomial Characters

• St. Andrews, August 2013

12 / 15

Main results

As a Corollary of Theorem B we get.

• ♣

❥

• ✭P✮ ✿ P❥

P ✷ ❙②❧♣✭●✮ ♣ ✤ ✷ ■rr✭●✮ ♣ ◗❥●❥♣ ✤ ❇✙ ❇✙

Carolina Vallejo (Universitat de Val` encia) Certain Monomial Characters

• St. Andrews, August 2013

13 / 15

Main results

As a Corollary of Theorem B we get.

Corollary C

Let ● be a ♣-solvable group. Suppose that ❥N●✭P✮ ✿ P❥ is odd, where P ✷ ❙②❧♣✭●✮ for some prime ♣. If ✤ ✷ ■rr✭●✮ has degree not divisible by ♣ and its field of values is contained in ◗❥●❥♣, then every subnormal constituent of ✤ is monomial. ❇✙ ❇✙

Carolina Vallejo (Universitat de Val` encia) Certain Monomial Characters

• St. Andrews, August 2013

13 / 15

Main results

As a Corollary of Theorem B we get.

Corollary C

Let ● be a ♣-solvable group. Suppose that ❥N●✭P✮ ✿ P❥ is odd, where P ✷ ❙②❧♣✭●✮ for some prime ♣. If ✤ ✷ ■rr✭●✮ has degree not divisible by ♣ and its field of values is contained in ◗❥●❥♣, then every subnormal constituent of ✤ is monomial. Key: Subnormal constituents of ❇✙ characters are ❇✙ characters.

Carolina Vallejo (Universitat de Val` encia) Certain Monomial Characters

• St. Andrews, August 2013

13 / 15

Main results

As a Corollary of Theorem B we get.

Corollary C

Let ● be a ♣-solvable group. Suppose that ❥N●✭P✮ ✿ P❥ is odd, where P ✷ ❙②❧♣✭●✮ for some prime ♣. If ✤ ✷ ■rr✭●✮ has degree not divisible by ♣ and its field of values is contained in ◗❥●❥♣, then every subnormal constituent of ✤ is monomial. Key: Subnormal constituents of ❇✙ characters are ❇✙ characters. Gow’s Theorem and Theorem A do not provide information about the subnormal constituents.

Carolina Vallejo (Universitat de Val` encia) Certain Monomial Characters

• St. Andrews, August 2013

13 / 15

Main results

We also obtain the following consequence.

• ♣

❥

• ✭P✮ ✿ P❥

P ✷ ❙②❧♣✭●✮ ♣ ♣ ◗❥●❥♣

• ✭P✮

P❂P ✵

Carolina Vallejo (Universitat de Val` encia) Certain Monomial Characters

• St. Andrews, August 2013

14 / 15

Main results

We also obtain the following consequence.

Corollary D

Let ● be a ♣-solvable group. Assume that ❥N●✭P✮ ✿ P❥ is odd, where P ✷ ❙②❧♣✭●✮ for some prime ♣. The number of irreducible characters which have degree not divisible by ♣ and field of values contained in ◗❥●❥♣ equals the number of orbits under the natural action of N●✭P✮ on P❂P ✵.

Carolina Vallejo (Universitat de Val` encia) Certain Monomial Characters

• St. Andrews, August 2013

14 / 15

Main results

We also obtain the following consequence.

Corollary D

Let ● be a ♣-solvable group. Assume that ❥N●✭P✮ ✿ P❥ is odd, where P ✷ ❙②❧♣✭●✮ for some prime ♣. The number of irreducible characters which have degree not divisible by ♣ and field of values contained in ◗❥●❥♣ equals the number of orbits under the natural action of N●✭P✮ on P❂P ✵. The number of such characters can be computed locally.

Carolina Vallejo (Universitat de Val` encia) Certain Monomial Characters

• St. Andrews, August 2013

14 / 15

Main results

Thanks for your attention!

Carolina Vallejo (Universitat de Val` encia) Certain Monomial Characters

• St. Andrews, August 2013

15 / 15