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Difference between 2 questions on the alternating group $A_4$.

Constructing the character table of a groupQuestions for compact lie group representations.Degrees of irreducible complex characters of alternating groupsOrder of group elements from a character tableDescribe all one-dimensional representations of the alternating group A4.About character table of S3Representations of $SL_2(mathbbF_3)$Intrinsic definition of group characterscharacter table of groupConstructing groups by combining irreducible representations

0

$begingroup$

I have to answer 2 questions on the alternating group $A_4$:

One of them asks me to find all irreducible complex representations of the group $A_4$ (and calculate their characters) while the other one asks me to find all 1-dimensional representations of $A_4$. I have answered the second question without using the character table. but I feel like the answer of the first question like the second one, could anyone explain the difference for me please?

Thank you!

representation-theory

share|cite|improve this question

asked yesterday

hopefullyhopefully

250114

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  I mean, yes, if you know all irreducible representations of $A_4$ then you certainly know the $1$-dimensional representations. Presumably if you do the first question first, then the second requires more work.
  $endgroup$
  – Elliot G
  yesterday
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  For those of us following from the peanut gallery who haven't had algebra in decades, would someone mind linking to a definition of, or explaining what is meant by, representations of a group? Thanks.
  $endgroup$
  – Robert Shore
  yesterday
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @RobertShore Of course. Please see here.
  $endgroup$
  – Dietrich Burde
  yesterday
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @DietrichBurde So it's an action of a group on a vector space that respects vector addition and scalar multiplication, which makes it a homomorphism into the group of automorphisms of the vector space? Thanks very much.
  $endgroup$
  – Robert Shore
  yesterday
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  If you have a character table of $S_4$ at hand you may want to check what happens when you restrict those to $A_4$. Indeed, the second exercise can be done without the first, and it may be more natural to do that first.
  $endgroup$
  – Jyrki Lahtonen
  yesterday
  

  
  
  
  

add a comment | 

0

$begingroup$

I have to answer 2 questions on the alternating group $A_4$:

One of them asks me to find all irreducible complex representations of the group $A_4$ (and calculate their characters) while the other one asks me to find all 1-dimensional representations of $A_4$. I have answered the second question without using the character table. but I feel like the answer of the first question like the second one, could anyone explain the difference for me please?

Thank you!

representation-theory

share|cite|improve this question

asked yesterday

hopefullyhopefully

250114

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  I mean, yes, if you know all irreducible representations of $A_4$ then you certainly know the $1$-dimensional representations. Presumably if you do the first question first, then the second requires more work.
  $endgroup$
  – Elliot G
  yesterday
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  For those of us following from the peanut gallery who haven't had algebra in decades, would someone mind linking to a definition of, or explaining what is meant by, representations of a group? Thanks.
  $endgroup$
  – Robert Shore
  yesterday
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @RobertShore Of course. Please see here.
  $endgroup$
  – Dietrich Burde
  yesterday
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @DietrichBurde So it's an action of a group on a vector space that respects vector addition and scalar multiplication, which makes it a homomorphism into the group of automorphisms of the vector space? Thanks very much.
  $endgroup$
  – Robert Shore
  yesterday
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  If you have a character table of $S_4$ at hand you may want to check what happens when you restrict those to $A_4$. Indeed, the second exercise can be done without the first, and it may be more natural to do that first.
  $endgroup$
  – Jyrki Lahtonen
  yesterday
  

  
  
  
  

add a comment | 

$begingroup$

I have to answer 2 questions on the alternating group $A_4$:

One of them asks me to find all irreducible complex representations of the group $A_4$ (and calculate their characters) while the other one asks me to find all 1-dimensional representations of $A_4$. I have answered the second question without using the character table. but I feel like the answer of the first question like the second one, could anyone explain the difference for me please?

Thank you!

representation-theory

share|cite|improve this question

asked yesterday

hopefullyhopefully

250114

$endgroup$

share|cite|improve this question

asked yesterday

hopefullyhopefully

250114

share|cite|improve this question

asked yesterday

hopefullyhopefully

250114

asked yesterday

hopefullyhopefully

250114

asked yesterday

hopefullyhopefully

250114