# Representation Theory of Finite Groups - CH 18 Dummit and Foote

• Math Amateur
In summary: This is not completely correct. A field is needed in order to define the operations of addition, multiplication, and comparison on the scalars. Without a field, the scalars would not be commutable.
Math Amateur
Gold Member
MHB
I am reading Dummit and Foote on Representation Theory CH 18

I am struggling with the following text on page 843 - see attachment and need some help.

The text I am referring to reads as follows - see attachment page 843 for details

$\phi ( g ) ( \alpha v + \beta w ) = g \cdot ( \alpha v + \beta w )$

$$= g \cdot ( \alpha v ) + g \cdot ( \beta w )$$

$$= \alpha ( g \cdot v ) + \beta ( g \cdot w )$$

$$= \alpha \phi ( g ) ( v ) + \beta \phi ( g ) (w)$$

Now my problem with the above concerns $$g \cdot ( \alpha v ) + g \cdot ( \beta w )$$ $$= \alpha ( g \cdot v ) + \beta ( g \cdot w )$$

This looks like it is just using the fact that elements of F commute with elements of g as in $$g \cdot ( \alpha v ) = \alpha ( g \cdot v )$$

BUT ... this is not just an element of F commuting with an element of G as in $$(1_F h ) ( \alpha 1_G) = ( \alpha 1_G ) ( 1_F h )$$ ...

the statement above involves the $$\cdot$$ operation which is (to quote D&F) " the given action of the ring element g on the element v of V" { why "ring" element? }

Doesn't the fact that we are dealing with an action mixed with terms like $$\alpha v$$ involving a field element multiplied by a vector complicate things ...

how do we formally and explicitly justify $$g \cdot ( \alpha v ) = \alpha ( g \cdot v )$$?

How do we justify taking $$\alpha$$ out through the the action $$\cdot$$ ? Why are we justified in doing this?

Peter

#### Attachments

• Dummit and Foote - Ch 18 - pages 840 - 843 - Representation Theory.pdf
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The group acts on the vector space by linear transformations. So the formulas just express linearity.

This just says that the vector space is a module over the group ring.

BTW:I do not see why these ideas require a representation of the group. Why can't the linear transformations be singular?

For instance take any linear transformation of a vector space. Then the vector space is a module of the group ring using exactly the same rules. In fact, in this case the vector space becomes a module over the ring of polynomials in one indeterminate.

Are you saying that if I go to the axioms of a vector space over a group ring (not sure where these are actually explicitly specified [I like to be very sure what I am doing is legitimate :-) ] then I will find justification in the axions

By the way it would help me a lot if you could explicitly justify $g \cdot ( \alpha v ) = \alpha ( g \cdot v )$ - perhaps explictly referring to the appropriate axioms of a module over a group ring.

Thanks for your help so far by the way!

Peter

Math Amateur said:
Are you saying that if I go to the axioms of a vector space over a group ring (not sure where these are actually explicitly specified [I like to be very sure what I am doing is legitimate :-) ] then I will find justification in the axions

By the way it would help me a lot if you could explicitly justify $g \cdot ( \alpha v ) = \alpha ( g \cdot v )$ - perhaps explictly referring to the appropriate axioms of a module over a group ring.

Thanks for your help so far by the way!

Peter

Certainly the relations you are asking about follow from linearity of the action of G on the vector space.

But the scalars can be thought of as elements of the group ring. Just multiply the scalar times the identity element of the group. Then linearity follows from the associativity of the action.

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Thanks

Peter

Math Amateur said:
Thanks

Peter

BTW: The idea of a group ring does not require a field. One can use any commuataive ring as scalars. A field is used in your case because the subject is representations of the group on a vector space.

Suppose though that you wanted to study representations of a group on a lattice such as the points in Euclidean space that have all integer coordinates. Then your scalars would have to be restricted to the integers. This integer group ring is also used to define the cohomology of a group.

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Hi Lavinia

You write "The idea of a group ring does not require a field." ... yes understand ..

Peter

Just to be sure ...

Are you saying that $$g \cdot ( \alpha v ) = \alpha ( g \cdot v )$$ follows from the axioms of a module over a group ring?

Peter

Math Amateur said:
Just to be sure ...

Are you saying that $$g \cdot ( \alpha v ) = \alpha ( g \cdot v )$$ follows from the axioms of a module over a group ring?

Peter

I think so because $\alpha$ = $\alpha$. [identity]. So the associative axiom of the group ring allows you to factor $\alpha$ out. Then use that the identity symbol in the group ring commutes with the other symbols.

In terms of the representations, you can think of scalar multiplication as multiplication by the scalar times the identity transformation.

## What is representation theory of finite groups?

Representation theory of finite groups is a branch of mathematics that studies the ways in which a finite group can be represented by matrices. It involves the study of linear representations of groups, where elements of the group are represented by invertible matrices, and group operations are represented by matrix multiplication.

## What are some applications of representation theory of finite groups?

Representation theory of finite groups has several applications in different fields of mathematics, including algebraic number theory, algebraic geometry, and combinatorics. It also has applications in physics, particularly in quantum mechanics and particle physics, where it is used to study symmetry properties of physical systems.

## What are the main tools used in representation theory of finite groups?

The main tools used in representation theory of finite groups include group theory, linear algebra, and character theory. Group theory provides the framework for studying group representations, while linear algebra is used to manipulate matrices and vector spaces. Character theory, on the other hand, is used to analyze the properties of group representations.

## What is the importance of irreducible representations in representation theory of finite groups?

Irreducible representations are an essential concept in representation theory of finite groups. They are representations that cannot be broken down into smaller representations, and they provide a unique way to classify representations. Irreducible representations also have many useful properties, such as orthogonality and completeness, which make them valuable tools in the study of group representations.

## What are some current research topics in representation theory of finite groups?

Some current research topics in representation theory of finite groups include the study of fusion categories, which are higher-dimensional generalizations of group representations, and the application of representation theory to other areas of mathematics, such as cryptography and coding theory. Other research areas include the study of modular representations, which are representations that arise from finite groups that have a non-trivial center.

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