Linear representation theory of Mathieu group:M11: Difference between revisions

Revision as of 01:59, 19 April 2012

This article gives specific information, namely, linear representation theory, about a particular group, namely: Mathieu group:M11.
View linear representation theory of particular groups | View other specific information about Mathieu group:M11

Summary

Item	Value
degrees of irreducible representations over a splitting field (such as ${\overline {\mathbb {Q} }}$ or $\mathbb {C}$ )	1,10,10,10,11,16,16,44,45,55 grouped form: 1 (1 time), 10 (3 times), 11 (1 time), 16 (2 times), 44 (1 time), 45 (1 time, 55 (1 time) maximum: 55, quasirandom degree: 10, number: 10, lcm: 7920, sum of squares: 7920

Family contexts

Family name	Parameter value	General discussion of linear representation theory of family
Mathieu group	degree 11, i.e., the group $M_{11}$	linear representation theory of Mathieu groups

GAP implementation

Degrees of irreducible representations

The degrees of irreducible representations can be computed using the CharacterDegrees, CharacterTable, and MathieuGroup functions:

gap> CharacterDegrees(CharacterTable(MathieuGroup(11)));
[ [ 1, 1 ], [ 10, 3 ], [ 11, 1 ], [ 16, 2 ], [ 44, 1 ], [ 45, 1 ], [ 55, 1 ] ]

Character table

The character table can be computed using the Irr, CharacterTable, and MathieuGroup functions:

gap> Irr(CharacterTable(MathieuGroup(11)));
[ Character( CharacterTable( Group([ (1,2,3,4,5,6,7,8,9,10,11), (3,7,11,8)(4,10,5,6) ]) ), [ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ] ),
  Character( CharacterTable( Group([ (1,2,3,4,5,6,7,8,9,10,11), (3,7,11,8)(4,10,5,6) ]) ), [ 10, 0, 0, 2, 2, 0, -1, 1, -1, -1 ] ),
  Character( CharacterTable( Group([ (1,2,3,4,5,6,7,8,9,10,11), (3,7,11,8)(4,10,5,6) ]) ), [ 10, -E(8)-E(8)^3, E(8)+E(8)^3, 0, -2, 0, 1, 1, -1, -1 ] ),
  Character( CharacterTable( Group([ (1,2,3,4,5,6,7,8,9,10,11), (3,7,11,8)(4,10,5,6) ]) ), [ 10, E(8)+E(8)^3, -E(8)-E(8)^3, 0, -2, 0, 1, 1, -1, -1 ] ),
  Character( CharacterTable( Group([ (1,2,3,4,5,6,7,8,9,10,11), (3,7,11,8)(4,10,5,6) ]) ), [ 11, -1, -1, -1, 3, 1, 0, 2, 0, 0 ] ),
  Character( CharacterTable( Group([ (1,2,3,4,5,6,7,8,9,10,11), (3,7,11,8)(4,10,5,6) ]) ),
    [ 16, 0, 0, 0, 0, 1, 0, -2, E(11)^2+E(11)^6+E(11)^7+E(11)^8+E(11)^10, E(11)+E(11)^3+E(11)^4+E(11)^5+E(11)^9 ] ), Character( CharacterTable( Group(
    [ (1,2,3,4,5,6,7,8,9,10,11), (3,7,11,8)(4,10,5,6) ]) ), [ 16, 0, 0, 0, 0, 1, 0, -2, E(11)+E(11)^3+E(11)^4+E(11)^5+E(11)^9,
      E(11)^2+E(11)^6+E(11)^7+E(11)^8+E(11)^10 ] ), Character( CharacterTable( Group([ (1,2,3,4,5,6,7,8,9,10,11), (3,7,11,8)(4,10,5,6) ]) ),
    [ 44, 0, 0, 0, 4, -1, 1, -1, 0, 0 ] ), Character( CharacterTable( Group([ (1,2,3,4,5,6,7,8,9,10,11), (3,7,11,8)(4,10,5,6) ]) ),
    [ 45, -1, -1, 1, -3, 0, 0, 0, 1, 1 ] ), Character( CharacterTable( Group([ (1,2,3,4,5,6,7,8,9,10,11), (3,7,11,8)(4,10,5,6) ]) ),
    [ 55, 1, 1, -1, -1, 0, -1, 1, 0, 0 ] ) ]

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 ! Item !! Value
 |-
-| [[degrees of irreducible representations]] over a [[splitting field]] (such as <math>\overline{\mathbb{Q}}</math> or <math>\mathbb{C}</math>) || 1,10,10,10,11,16,16,44,45,55<br>grouped form: 1 (1 time), 10 (3 times), 11 (1 time), 16 (2 times), 44 (1 time), 45 (1 time, 55 (1 time)<br>[[maximum degree of irreducible representation|maximum]]: 55, [[number of irreducible representations equals number of conjugacy classes|number]]: 10, [[sum of squares of degrees of irreducible representations equals order of group|sum of squares]]: 7920
+| [[degrees of irreducible representations]] over a [[splitting field]] (such as <math>\overline{\mathbb{Q}}</math> or <math>\mathbb{C}</math>) || 1,10,10,10,11,16,16,44,45,55<br>grouped form: 1 (1 time), 10 (3 times), 11 (1 time), 16 (2 times), 44 (1 time), 45 (1 time, 55 (1 time)<br>[[maximum degree of irreducible representation|maximum]]: 55, [[quasirandom degree]]: 10, [[number of irreducible representations equals number of conjugacy classes|number]]: 10, [[lcm of degrees of irreducible representations|lcm]]: 7920, [[sum of squares of degrees of irreducible representations equals order of group|sum of squares]]: 7920
 |}
 <section end="summary"/>