Maximum degree of irreducible representation

This term is related to: linear representation theory
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This article defines an arithmetic function on groups
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Definition

For a group over a field

Suppose $G$ is a group and $K$ is a field. The maximum degree of irreducible representation of $G$ is defined as the maximum of all the degrees of irreducible representations of $G$ over $K$.

Typical context: finite group and splitting field

The typical context is where $G$ is a finite group and $K$ is a splitting field for $G$. In particular, the characteristic of $K$ is either zero or is a prime not dividing the order of $G$, and every irreducible representation of $G$ over any extension field of $K$ can be realized over $K$.

Note that the maximum degree of irreducible representation depends (if at all) only on the characteristic of the field $K$. This is because the degrees of irreducible representations over a splitting field depend only on the characteristic of the field.

Default case: characteristic zero

By default, when referring to the maximum degree of irreducible representation, we refer to the case of characteristic zero, and we can in particular take $K=\mathbb {C}$.

Facts

Subgroups

• Maximum degree of irreducible representation of subgroup is less than or equal to maximum degree of irreducible representation of whole group
• Maximum degree of irreducible representation of group is less than or equal to product of maximum degree of irreducible representation of subgroup and index of subgroup

Quotients and direct products

• Maximum degree of irreducible representation of quotient group is less than or equal to maximum degree of irreducible representation of whole group
• Maximum degree of irreducible representation of direct product is maximum of maximum degrees of irreducible representation of each direct factor

Field changes

• Maximum degree of irreducible real representation is at most twice maximum degree of irreducible complex representation

Particular cases

For any finite abelian group, all the irreducible representations over a splitting field are one-dimensional, so the maximum degree of irreducible representation over any splitting field is one-dimensional. The situation may be different over non-splitting fields.

Group	Order	Second part of GAP ID	Maximum degree of irreducible representation over $\mathbb {C}$	Maximum degree of irreducible representation over $\mathbb {R}$	Maximum degree of irreducible representation over $\mathbb {Q}$	General note on degrees of irreducible representations
trivial group	1	1	1	1	1	always 1, regardless of the field
cyclic group:Z2	2	1	1	1	1	always 1, any field of characteristic not 2
cyclic group:Z3	3	1	1	2	2	either 1 or 2, depending on whether the field is a splitting field
cyclic group:Z4	4	1	1	2	2	either 1 or 2, depending on whether the field is a splitting field
Klein four-group	4	2	1	1	1	always 1, any field of characteristic not 2
cyclic group:Z5	5	1	1	2	4	1, 2, or 4, depending on how $x^{4}+x^{3}+x^{2}+x+1$ splits in the field