Quasirandom degree

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This article defines an arithmetic function on groups
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Definition

Definition for finite groups

Suppose G is a finite group. The quasirandom degree of G is defined in the following equivalent ways:

  1. It is the minimum possible degree of a nontrivial linear representation of G over the field of complex numbers.
  2. It is the minimum possible degree of an nontrivial irreducible linear representation of G over the field of complex numbers (i.e., it is the smallest of the degrees of irreducible representations once we throw out the trivial representation).
  3. It is the minimum of the degrees of nontrivial linear representations of G over all possible fields of characteristic zero.
  4. It is the minimum possible degree of an nontrivial irreducible linear representation of G over all possible fields of characteristic zero.

Note that for the trivial group, the quasirandom degree is taken to be +\infty.

We say that a nontrivial finite group G is d-quasirandom if the quasirandom degree of G is at least d, i.e., if every nontrivial irreducible linear representation of G has degree at least d.

Definition for locally compact groups

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Related notions

Facts

Quasirandom degree of one

Relation with subgroups and quotients

External links