Representation of finite group question

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Discussion Overview

The discussion revolves around the properties of irreducible representations of finite groups, specifically focusing on proving that any irreducible representation has a degree at most equal to the order of the group. Participants explore related concepts, such as the reducibility of representations and the implications of group algebras in this context.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant questions how to prove that any irreducible representation of a finite group G has degree at most |G|, suggesting that representations of degree greater than |G| are reductible.
  • Another participant expresses suspicion that the conjecture may be true, noting that the set Gv should span a direct summand of the entire G-vector space, but admits to not having proven it yet.
  • A participant discusses the group algebra FG and the associated FG-module V, arguing that the dimension of the vector subspace W = span(Gv) is at most |G|, leading to the conclusion that |G| must be greater than or equal to dim(V) if V is irreducible.
  • One participant reflects on their misunderstanding regarding direct sum decompositions and suggests proving that FG is itself a reducible representation.
  • Another participant posits that the |G|-dimensional FG-module FG is reducible, providing an example with a specific vector v and questioning whether every |G|-dimensional FG-module is isomorphic to FG.
  • There is a discussion about the implications of restricting the representation to a cyclic subgroup, noting that it results in a direct sum of 1-dimensional representations, which must be fewer than the order of G if the representation of the entire group is irreducible.

Areas of Agreement / Disagreement

Participants express varying levels of certainty regarding the conjectures and proofs presented. There is no clear consensus on the sufficiency of certain arguments or the implications of the properties discussed, indicating ongoing debate and exploration of the topic.

Contextual Notes

Some participants reference specific exercises from texts, indicating that the discussion may depend on definitions and interpretations of irreducibility and reducibility in the context of group representations.

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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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.
 
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|!
 
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?
 
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?
 
quasar987 said:
But why do you think this suffices?
I thought it works as an addendum to the previous proof.
 
In what sense?
 
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.
 
Last edited:
quasar987 said:
In what sense?
That it gives more information about the span of Gv.
 

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