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

From Groupprops

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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]] | ||

+ | - <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