Quasirandom degree

This article defines an arithmetic function on groups
View other such arithmetic functions

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

• Maximum degree of irreducible representation is the maximum of the degrees of irreducible representations

Facts

Quasirandom degree of one

• For any nontrivial finite abelian group, the quasirandom degree is 1. This follows from the fact that abelian implies every irreducible representation is one-dimensional
• More generally, the quasirandom degree of a group is greater than 1 if and only if the group is a perfect group. This follows from the fact that number of one-dimensional representations equals order of abelianization, so if the abelianization is nontrivial, there is more than one one-dimensional representation, and hence a nontrivial one-dimensional representation.

Relation with subgroups and quotients

• Quasirandom degree of quotient group is bounded below by quasirandom degree of whole group
• Quasirandom degree of extension group is bounded below by minimum of quasirandom degrees of normal subgroup and quotient group
• Quasirandom degree of group is bounded below by minimum of quasirandom degrees of generating subgroups

External links

• Blog post by Terence Tao on quasirandom groups (he does not use quasirandom degree but uses the $d$-quasirandom notation).