Difference between revisions of "Quiz:Degrees of irreducible representations"

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<quiz display=simple>
 
<quiz display=simple>
{Which of the following is ''not'' a possibility for the multiset of the [[degrees of irreducible representations]] of a finite group over a splitting field of characteristic zero?
+
{Which of the following is ''not'' a possibility for the multiset of the [[degrees of irreducible representations]] of a finite group over a [[splitting field]] of characteristic zero (such as the complex numbers)?
 
|type="()"}
 
|type="()"}
 
- 1,1  
 
- 1,1  
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+ None of the above, i.e., they are all possibilities  
 
+ None of the above, i.e., they are all possibilities  
 
|| ''Right''. (1,1) arises for [[cyclic group:Z2]], the others all arise for the [[general affine group of degree one]] <math>GA(1,q)</math> for <math>q = 3,4,5</math> respectively. See [[linear representation theory of general affine group of degree one over a finite field]].
 
|| ''Right''. (1,1) arises for [[cyclic group:Z2]], the others all arise for the [[general affine group of degree one]] <math>GA(1,q)</math> for <math>q = 3,4,5</math> respectively. See [[linear representation theory of general affine group of degree one over a finite field]].
 +
 +
{What is the largest possible value of the [[maximum degree of irreducible representation]] for a group of order 24 over a [[splitting field]] of characteristic zero (such as the complex numbers)?
 +
|type="()"}
 +
- 2
 +
+ 3
 +
|| See [[linear representation theory of groups of order 24]].
 +
- 4
 +
- 6
 +
- 8
 +
 +
{What is the largest possible value of the [[maximum degree of irreducible representation]] for a group of order <math>2^{2n + 1}</math> over a [[splitting field]] of characteristic zero (such as the field of complex numbers) where <math>n</math> is a positive integer?
 +
|type="()"}
 +
- 2
 +
+ <math>2^n</math>
 +
|| The maximum occurs for extraspecial groups, see [[linear representation theory of extraspecial groups]]. Obtaining this as an upper bound is easy: see [[order of inner automorphism group bounds square of degree of irreducible representation]], and [[prime power order implies not centerless]]
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- <math>2^{n + 1}</math>
 +
- <math>2^{2n - 1}</math>
 +
- <math>2^{2n}</math>
 +
- <math>2^{2n + 1}</math>
 
</quiz>
 
</quiz>

Revision as of 16:44, 3 August 2011

1 Which of the following is not a possibility for the multiset of the degrees of irreducible representations of a finite group over a splitting field of characteristic zero (such as the complex numbers)?

1,1
1,1,2
1,1,1,3
1,1,1,1,4
None of the above, i.e., they are all possibilities

2 What is the largest possible value of the maximum degree of irreducible representation for a group of order 24 over a splitting field of characteristic zero (such as the complex numbers)?

2
3
4
6
8

3 What is the largest possible value of the maximum degree of irreducible representation for a group of order 2^{2n + 1} over a splitting field of characteristic zero (such as the field of complex numbers) where n is a positive integer?

2
2^n
2^{n + 1}
2^{2n - 1}
2^{2n}
2^{2n + 1}