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

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

− | - 1,1,2 | + | - 1,1,1,1,2 |

− | - 1,1,1,3 | + | || ''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, | + | + 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="()"} | ||

+ | - 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]] <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> | ||

+ | |||

+ | ==Maximum and divisibility== | ||

+ | |||

+ | For all the questions below, we consider irreducible representations over a splitting field of characteristic zero, such as the field of complex numbers. | ||

+ | |||

+ | <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)? | ||

+ | |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]] | ||

+ | - <math>2^{n + 1}</math> | ||

+ | - <math>2^{2n - 1}</math> | ||

+ | - <math>2^{2n}</math> | ||

+ | - <math>2^{2n + 1}</math> | ||

+ | |||

+ | {It is true in general that [[degree of irreducible representation divides index of abelian normal subgroup]], when we are working with irreducible representations of finite groups over splitting fields of characteristic zero. Which of the following gives an example of a group where the least common multiple of the degrees of irreducible representations is ''strictly'' smaller than the greatest common divisor of the index values of all abelian normal subgroups? | ||

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

+ | - [[symmetric group:S3]] | ||

+ | - [[symmetric group:S4]] | ||

+ | - [[alternating group:A4]] | ||

+ | + [[special linear group:SL(2,3)]] | ||

+ | - [[alternating group:A5]] | ||

+ | - [[special linear group:SL(2,5)]] | ||

+ | </quiz> | ||

+ | |||

+ | ==Groups of prime power order: smallest counterexamples== | ||

+ | |||

+ | <quiz display=simple> | ||

+ | {What is the ''smallest'' power <math>2^n</math> such that there exist two groups of order <math>2^n</math> of the same [[nilpotency class]] but with different multisets of [[degrees of irreducible representations]] over the field of complex numbers? | ||

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

+ | - 8 | ||

+ | - 16 | ||

+ | + 32 | ||

+ | || See [[degrees of irreducible representations need not determine nilpotency class]] | ||

+ | - 64 | ||

+ | - 128 | ||

+ | |||

+ | {What is the ''smallest'' power <math>2^n</math> such that there exist two groups of order <math>2^n</math> with the same multiset of [[degrees of irreducible representations]] over the field of complex numbers but such that the [[conjugacy class size statistics of a finite group|conjugacy class size statistics]] are different for the groups? | ||

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

+ | - 8 | ||

+ | || See [[[[linear representation theory of groups of order 8#Degrees of irreducible representations]] | ||

+ | - 16 | ||

+ | || See [[linear representation theory of groups of order 16#Degrees of irreducible representations]] | ||

+ | - 32 | ||

+ | || See [[linear representation theory of groups of order 32#Degrees of irreducible representations]] | ||

+ | + 64 | ||

+ | || See [[degrees of irreducible representations need not determine conjugacy class size statistics]]. See also [[linear representation theory of groups of order 64#Degrees of irreducible representations]] | ||

+ | - 128 | ||

+ | || See [[[[linear representation theory of groups of order 128#Degrees of irreducible representations]] | ||

+ | |||

+ | {What is the ''smallest'' power <math>2^n</math> such that there exist two groups of order <math>2^n</math> with the [[conjugacy class size statistics of a finite group|conjugacy class size statistics]] but such that the [[degrees of irreducible representations]] are different for the groups? | ||

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

+ | - 8 | ||

+ | || See [[linear representation theory of groups of order 8#Degrees of irreducible representations]] | ||

+ | - 16 | ||

+ | || See [[linear representation theory of groups of order 16#Degrees of irreducible representations]] | ||

+ | - 32 | ||

+ | || See [[linear representation theory of groups of order 32#Degrees of irreducible representations]] | ||

+ | - 64 | ||

+ | || See [[linear representation theory of groups of order 64#Degrees of irreducible representations]] | ||

+ | + 128 | ||

+ | || See [[conjugacy class size statistics need not determine degrees of irreducible representations]], see also [[linear representation theory of groups of order 128#Degrees of irreducible representations]] | ||

</quiz> | </quiz> |

## Latest revision as of 17:33, 3 August 2011

## Find the feasible or infeasible degrees of irreducible representations

## Maximum and divisibility

For all the questions below, we consider irreducible representations over a splitting field of characteristic zero, such as the field of complex numbers.

## Groups of prime power order: smallest counterexamples