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

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{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?
 
|type="()"}
 
|type="()"}
- 1,1
+
- 1,1 || ''Wrong''. This arises as the degrees of irreducible representations for [[cyclic group:Z2]]. See [[linear representation theory of cyclic group:Z2]].
- 1,1,2
+
- 1,1,2 || ''Wrong''. This arises as the degrees of irreducible representations for [[symmetric group:S3]], which can alternatively be viewed as the [[general affine group of degree one]] over [[field:F3]]. See [[linear representation theory of symmetric group:S3]].
- 1,1,1,3
+
- 1,1,1,3 || ''Wrong''. This arises as the degrees of irreducible representations for [[alternating group:A4]], which can alternatively be viewed as the [[general affine group of degree one]] over [[field:F4]]. See [[linear representation theory of alternating group:A4]].
- 1,1,1,1,4
+
- 1,1,1,1,4 || ''Wrong''. This arises as the degrees of irreducible representations of [[GA(1,5)]], the [[general affine group of degree one]] over [[field:F5]].
+ 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]].
 
</quiz>
 
</quiz>

Revision as of 16:38, 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?

1,1 || Wrong. This arises as the degrees of irreducible representations for cyclic group:Z2. See linear representation theory of cyclic group:Z2.
1,1,2 || Wrong. This arises as the degrees of irreducible representations for symmetric group:S3, which can alternatively be viewed as the general affine group of degree one over field:F3. See linear representation theory of symmetric group:S3.
1,1,1,3 || Wrong. This arises as the degrees of irreducible representations for alternating group:A4, which can alternatively be viewed as the general affine group of degree one over field:F4. See linear representation theory of alternating group:A4.
1,1,1,1,4 || Wrong. This arises as the degrees of irreducible representations of GA(1,5), the general affine group of degree one over field:F5.
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 GA(1,q) for q = 3,4,5 respectively. See linear representation theory of general affine group of degree one over a finite field.