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

From Groupprops

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+ | ==Find the feasible or infeasible degrees of irreducible representations== | ||

+ | |||

<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 (such as the complex numbers)? | ||

+ | |type="()"} | ||

+ | - 1,1,1,1,2 | ||

+ | || ''Wrong''. This occurs for both [[dihedral group:D8]] (see [[linear representation theory of dihedral group:D8]] and [[quaternion group]] (see [[linear representation theory of quaternion group]]). See also [[linear representation theory of groups of order 8]]. | ||

+ | + 1,1,1,1,3 | ||

+ | || ''Right''. This is impossible because, by [[sum of squares of degrees of irreducible representations equals order of group]], the order of the group is 13. But a group of order 13 must be cyclic of prime order, and hence must have all its degrees of irreducible representations equal to 1. | ||

+ | - 1,1,1,1,4 | ||

+ | || ''Wrong''. This occurs for [[GA(1,5)]], the [[general affine group of degree one]] over [[field:F5]]. | ||

+ | - None of the above, i.e., they are all possibilities for the multiset of degrees of irreducible representations. | ||

+ | |||

+ | {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="()"} | ||

+ | - 1,1,2,3,3 | ||

+ | || ''Wrong''. This occurs as the degrees of irreducible representations for [[symmetric group:S4]] (a group of order 24). See [[linear representation theory of symmetric group:S4]]. See also [[linear representation theory of groups of order 24]]. | ||

+ | - 1,1,1,2,2,2,3 | ||

+ | || ''Wrong''. This occurs as the degrees of irreducible representations for [[special linear group:SL(2,3)]] (a group of order 24). See [[linear representation theory of special linear group:SL(2,3)]]. See also [[linear representation theory of groups of order 24]]. | ||

+ | - 1,1,1,1,2,2,2,2,2 | ||

+ | || ''Wrong''. This occurs as the degrees of irreducible representations for [[SmallGroup(24,8)]], a group of order 24. See also [[linear representation theory of groups of order 24]]. | ||

+ | + 1,1,1,1,1,1,1,2,2,3 | ||

+ | || ''Right''. By [[sum of squares of degrees of irreducible representations equals order of group]], the group has order 24. Further, we must have [[number of one-dimensional representations equals order of abelianization]], which must divide 24. However, here, the number of 1s is 7, which does not divide 24. Thus, this does not occur as the degrees of irreducible representations of a finite group. See also [[linear representation theory of groups of order 24]]. | ||

+ | - 1,1,1,1,1,1,1,1,2,2,2,2 | ||

+ | || ''Wrong''. This occurs as the degrees of irreducible representations of [[direct product of S3 and V4]], a group of order 24. See also [[linear representation theory of groups of order 24]]. | ||

{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="()"} | ||

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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]]. | ||

+ | </quiz> | ||

+ | |||

+ | <quiz display=simple> | ||

+ | {Which of the following statements is ''false'' in general about the [[degrees of irreducible representations]] of a finite group over a [[splitting field]] of characteristic zero? | ||

+ | |type="()"} | ||

+ | + The degree of any irreducible representation divides the index of any abelian subgroup in the group. | ||

+ | || ''Right''. See [[degree of irreducible representation need not divide index of abelian subgroup]] | ||

+ | - The degree of any irreducible representation is bounded by, but need not divide, the index of any abelian subgroup in the group. | ||

+ | || ''Wrong''. See [[index of abelian subgroup bounds degree of irreducible representation]]. | ||

+ | - The degree of any irreducible representation divides the index of any [[abelian normal subgroup]] in the group. | ||

+ | || ''Wrong''. See [[degree of irreducible representation divides index of abelian normal subgroup]]. | ||

{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)? | {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)? |

## Revision as of 16:59, 3 August 2011

## Find the feasible or infeasible degrees of irreducible representations