Maximum degree of irreducible representation: Difference between revisions

Latest revision as of 19:15, 3 May 2015

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 $K=\mathbb {C}$ .

Related notions

Quasirandom degree is the minimum of the degrees of nontrivial irreducible representations.

Facts

Subgroups

The proofs presented for these facts seem to rely on the assumption that the characteristic of the field does not divide the order of the group, although it might be possible to adapt them to the modular case:

Quotients and direct products

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, regardless of the field
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, regardless of the field
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

@@ Line 1: / Line 1: @@
 {{term related to|linear representation theory}}
-{arithmetic function on groups}}
+{{arithmetic function on groups}}
 ==Definition==
@@ Line 7: / Line 7: @@
 ===For a group over a field===
-Suppose <math>G</math> is a [[group]] and <math>K</math> is a [[field]]. The '''lcm of degrees of irreducible representations''' of <math>G</math> is defined as the maximum of all the [[defining ingredient::degrees of irreducible representations]] of <math>G</math> over <math>K</math>.
+Suppose <math>G</math> is a [[group]] and <math>K</math> is a [[field]]. The '''maximum degree of irreducible representation''' of <math>G</math> is defined as the maximum of all the [[defining ingredient::degrees of irreducible representations]] of <math>G</math> over <math>K</math>.
 ===Typical context: finite group and splitting field===
@@ Line 13: / Line 13: @@
 The typical context is where <math>G</math> is a [[finite group]] and <math>K</math> is a [[splitting field]] for <math>G</math>. In particular, the characteristic of <math>K</math> is either zero or is a prime not dividing the order of <math>G</math>, and every irreducible representation of <math>G</math> over any extension field of <math>K</math> can be realized over <math>K</math>.
-Note that the lcm of degrees of irreducible representations depends (if at all) only on the characteristic of the field <math>K</math>. This is because the [[degrees of irreducible representations]] over a splitting field depend only on the characteristic of the field.
+Note that the maximum degree of irreducible representation depends (if at all) only on the characteristic of the field <math>K</math>. 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 lcm of degrees of irreducible representations, we refer to the case of characteristic zero, and we can in particular take <math>K = \mathbb{C}</math>.
+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 <math>K = \mathbb{C}</math>.
+==Related notions==
+* [[Quasirandom degree]] is the minimum of the degrees of nontrivial irreducible representations.
+==Facts==
+===Subgroups===
+The proofs presented for these facts seem to rely on the assumption that the characteristic of the field does not divide the order of the group, although it might be possible to adapt them to the modular case:
+* [[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]]
+* In the case that we are over a [[splitting field]], we also have that [[degree of irreducible representation is bounded by index of abelian subgroup]] and [[order of inner automorphism group bounds square of degree of irreducible representation]]
+===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.
+{| class="sortable" border="1"
+! Group !! Order !! Second part of GAP ID !! Maximum degree of irreducible representation over <math>\mathbb{C}</math> !! Maximum degree of irreducible representation over <math>\R</math> !! Maximum degree of irreducible representation over <math>\mathbb{Q}</math> !! 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, regardless of the field
+|-
+| [[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, regardless of the field
+|-
+| [[cyclic group:Z5]] || 5 || 1 || 1 || 2 || 4 || 1, 2, or 4, depending on how <math>x^4 + x^3 + x^2 + x + 1</math> splits in the field
+|}