Linear representation theory of alternating group:A7: Difference between revisions

Latest revision as of 01:17, 19 April 2012

This article gives specific information, namely, linear representation theory, about a particular group, namely: alternating group:A7.
View linear representation theory of particular groups | View other specific information about alternating group:A7

Summary

Item	Value
degrees of irreducible representations over a splitting field	1,6,10,10,14,14,15,21,35 grouped form: 1 (1 time), 6 (1 time), 10 (2 times), 14 (2 times), 15 (1 time), 21 (1 time), 35 (1 time) maximum: 35, lcm: 210, number: 9, sum of squares: 2520
Minimal splitting field, i.e., field of realization of all irreducible representations (characteristic zero)	$Q (ω + ω^{2} + ω^{4})$ where $ω$ is a primitive seventh root of unity. This is the same as $Q (\sqrt{- 7})$ Quadratic extension of $Q$ Same as field generated by character values
Condition for a field of characteristic not 2,3,5, or 7, to be a splitting field	-7 should be a square in the field.
Minimal splitting field, i.e., field of realization of all irreducible representations in prime characteristic $p \neq 2, 3, 5, 7$	Case $p \equiv 1, 2, 4 (\mod 7)$ : prime field $F_{p}$ Case $p \equiv 3, 5, 6 (\mod 7)$ : quadratic extension $F_{p^{2}}$
Smallest size splitting field	field:F11

Family contexts

Family name	Parameter values	General discussion of linear representation theory of family
alternating group $A_{n}$	degree $n = 7$ , i.e., the group $A_{7}$	linear representation theory of alternating groups

GAP implementation

Degrees of irreducible representations

The degrees of irreducible representations can be computed using GAP's CharacterDegrees, CharacterTable, and AlternatingGroup functions:

gap> CharacterDegrees(AlternatingGroup(7));
[ [ 1, 1 ], [ 6, 1 ], [ 10, 2 ], [ 14, 2 ], [ 15, 1 ], [ 21, 1 ], [ 35, 1 ] ]

This means there is 1 degree 1 irreducible, 1 degree 6 irreducible, 2 degree 10 irreducibles, 2 degree 14 irreducibles, and 1 irreducible each of degrees 15, 21, 35.

Character table

The characters of irreducible representations can be computed using the Irr, CharacterTable, and CharacterDegree functions:

gap> Irr(CharacterTable(AlternatingGroup(7)));
[ Character( CharacterTable( Alt( [ 1 .. 7 ] ) ), [ 1, 1, 1, 1, 1, 1, 1, 1, 1 ] ), Character( CharacterTable( Alt( [ 1 .. 7 ] ) ),
    [ 6, 2, 3, -1, 0, 0, 1, -1, -1 ] ), Character( CharacterTable( Alt( [ 1 .. 7 ] ) ), [ 10, -2, 1, 1, 1, 0, 0, E(7)^3+E(7)^5+E(7)^6, E(7)+E(7)^2+E(7)^4
     ] ), Character( CharacterTable( Alt( [ 1 .. 7 ] ) ), [ 10, -2, 1, 1, 1, 0, 0, E(7)+E(7)^2+E(7)^4, E(7)^3+E(7)^5+E(7)^6 ] ),
  Character( CharacterTable( Alt( [ 1 .. 7 ] ) ), [ 14, 2, 2, 2, -1, 0, -1, 0, 0 ] ), Character( CharacterTable( Alt( [ 1 .. 7 ] ) ),
    [ 14, 2, -1, -1, 2, 0, -1, 0, 0 ] ), Character( CharacterTable( Alt( [ 1 .. 7 ] ) ), [ 15, -1, 3, -1, 0, -1, 0, 1, 1 ] ),
  Character( CharacterTable( Alt( [ 1 .. 7 ] ) ), [ 21, 1, -3, 1, 0, -1, 1, 0, 0 ] ), Character( CharacterTable( Alt( [ 1 .. 7 ] ) ),
    [ 35, -1, -1, -1, -1, 1, 0, 0, 0 ] ) ]

@@ Line 11: / Line 11: @@
 |-
 | [[degrees of irreducible representations]] over a [[splitting field]] || 1,6,10,10,14,14,15,21,35<br>grouped form: 1 (1 time), 6 (1 time), 10 (2 times), 14 (2 times), 15 (1 time), 21 (1 time), 35 (1 time)<br>[[maximum degree of irreducible representation|maximum]]: 35, [[lcm of degrees of irreducible representations|lcm]]: 210, [[number of irreducible representations equals number of conjugacy classes|number]]: 9, [[sum of squares of degrees of irreducible representations equals group order|sum of squares]]: 2520
+|-
+| [[Minimal splitting field]], i.e., field of realization of all irreducible representations (characteristic zero) || <math>\mathbb{Q}(\omega + \omega^2 + \omega^4)</math> where <math>\omega</math> is a primitive seventh root of unity. This is the same as <math>\mathbb{Q}(\sqrt{-7})</math><br>Quadratic extension of <math>\mathbb{Q}</math><br>Same as [[field generated by character values]]
+|-
+| Condition for a field of characteristic not 2,3,5, or 7, to be a splitting field || -7 should be a square in the field.
+|-
+| [[Minimal splitting field]], i.e., field of realization of all irreducible representations in prime characteristic <math>p \ne 2,3,5,7</math> || Case <math>p \equiv 1,2,4 \pmod 7</math>: prime field <math>\mathbb{F}_p</math><br>Case <math>p \equiv 3,5,6 \pmod 7</math>: quadratic extension <math>\mathbb{F}_{p^2}</math>
+|-
+| Smallest size splitting field || [[field:F11]]
 |}
 <section end="summary"/>
+==Family contexts==
+{| class="sortable" border="1"
+! Family name !! Parameter values !! General discussion of linear representation theory of family
+|-
+| [[alternating group]] <math>A_n</math> || degree <math>n = 7</math>, i.e., the group <math>A_7</math> || [[linear representation theory of alternating groups]]
+|}
 ==GAP implementation==
-The degrees of irreducible representations can be computed using GAP's [[GAP:CharacterDegrees|CharacterDegrees]] and [[GAP:AlternatingGroup|AlternatingGroup]] functions:
+===Degrees of irreducible representations===
+The degrees of irreducible representations can be computed using GAP's [[GAP:CharacterDegrees|CharacterDegrees]], [[GAP:CharacterTable|CharacterTable]], and [[GAP:AlternatingGroup|AlternatingGroup]] functions:
 <pre>gap> CharacterDegrees(AlternatingGroup(7));
@@ Line 23: / Line 41: @@
 This means there is 1 degree 1 irreducible, 1 degree 6 irreducible, 2 degree 10 irreducibles, 2 degree 14 irreducibles, and 1 irreducible each of degrees 15, 21, 35.
-The characters of irreducible representations can be computed using the [[GAP:CharacterTable|CharacterTable]] function:
+===Character table===
+The characters of irreducible representations can be computed using the [[GAP:Irr|Irr]], [[GAP:CharacterTable|CharacterTable]], and [[GAP:CharacterDegree|CharacterDegree]] functions:
 <pre>gap> Irr(CharacterTable(AlternatingGroup(7)));