Representation of finite group question

In summary, it is possible to prove that any irreducible representation of a finite group G has degree at most |G|, or equivalently, that every representation of degree greater than |G| is reducible. This can be shown by considering the vector subspace W = span(Gv) for any non-zero v in the FG-module associated with the representation. Additionally, it is possible to show that an irreducible representation has dimension strictly less than |G| by proving that FG, the |G|-dimensional FG-module, is itself reducible. This can be done by considering the restriction of the representation to a cyclic subgroup, which is a direct sum of 1-dimensional representations. Overall, this information provides insight into
  • #1
quasar987
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Does anyone know how to prove that any irreducible representation of a finite group G has degree at most |G|?

Equivalently, that every representation of degree >|G| is reductible.

Thx!
 
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  • #2
I suspect your conjecture is true -- for any vector v, the set Gv should span a direct summand of the entire G-vector space. But I haven't proven it yet, so grain of salt.
 
  • #3
If F is the field, write FG for the group algebra and call V the FG-module associated with a given representation of G. For any non zero v in V, Gv has at most |G| elements, and so the vector subspace W = span(Gv) has dimension at most |G| and it is clearly stable under the action of G (i.e., it is an FG-submodule of V). But W is non trivial and so if V is irreducible, it must be that W=V. Thus |G|>=dim(W)=dim(V).

I think this work, but according to Dummit & Foote Exercice 5 in the section on representation theory, we can do better and show that an irreducible representation has dimension strictly less than |G|!
 
  • #4
Oh! Silly me, I was looking for a direct sum decomposition -- I should have paid more attention to the definitions.


So... I think all we need to do now is to prove that FG is itself a reducible representation, right?
 
  • #5
It seems to me that the |G|-dimensional FG-module FG is reducible because if G={e,g_1,...g_r}, for v:=e+g_1+...g_r, we have that span(v) is a one dimensional FG-submodule of FG.

But why do you think this suffices? Is every |G|-dimensional FG-module isomorphic to FG?
 
  • #6
quasar987 said:
But why do you think this suffices?
I thought it works as an addendum to the previous proof.
 
  • #7
In what sense?
 
  • #8
If the field is the complex numbers:

The restriction of the representation to a cyclic subgroup is a direct sum of 1 dimensional representations. Since the representation of the entire group is ireducible the number of these 1 dimensional representations in the decomposition must be less that the order of G.
 
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  • #9
quasar987 said:
In what sense?
That it gives more information about the span of Gv.
 

What is a finite group?

A finite group is a mathematical structure that consists of a set of elements and a binary operation that combines any two elements to produce a third element. The set of elements is finite, meaning it has a specific number of elements, and the binary operation satisfies certain properties, such as closure, associativity, and identity.

What is the representation of a finite group?

The representation of a finite group is a way of describing the group's elements and operations using matrices or linear transformations. This allows us to study the group's properties and behavior in a more concrete and manageable way.

Why is the representation of a finite group important?

The representation of a finite group is important because it allows us to study and understand the group's structure, properties, and behavior in a more tangible way. It also has applications in various fields, such as physics, chemistry, and computer science.

How do you construct a representation of a finite group?

There are several methods for constructing a representation of a finite group, such as the Cayley table method, the character table method, and the permutation representation method. Each method has its advantages and is suitable for different types of groups.

What is the relationship between a group's representation and its subgroups?

The representation of a finite group and its subgroups are closely related. In fact, the representation theory of finite groups provides insights into the structure and properties of subgroups. For example, a subgroup can be described as a subset of the group's elements that are preserved by the group's representation.

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